


























































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































<'v ■>■'■'»■♦ 

' 


rS#"* 




S* 


‘V 

•1 


t 

■j 


i‘' 

S'\' 



declassiftrd 

authority Secretary of 


SEP 7 1960 

DfXonse memo 2 Auguat 1960 
LIBRAKY OF CONGRESS 


LC REGULATION: BEFORE SERVICING 
(5inSEpROl)UC'lN(I ANY PART OF TH: 
document, all classierA'HON 
mabkiNOS must be CANCfelMB:. 



. Librafy of Congress 










declassified 

By authority Secretary of 


SEP 7 1960 

Defense memo 2 August 1960 
LIBRARY OF CONGRESS 


SUMMARY TECHNICAL REPORT 
OF THE 

NATIONAL DEFENSE RESEARCH COMMITTEE 


LC REGULATION: BEFORE SERVICING 
OR REPRODUCING ANY PART OF THIS 
DOCUMENT, ALL CLASSIFICATION 
MARKINGS MUST BE CANCm:^ 


This document contains information affecting the national defense of 
the United States within the meaning of the Espionage Act, 50 U. S. C., 
31 and 32, as amended. Its transmission or the revelation of its contents 
in any manner to an unauthorized person is prohibited by law. 

This volume is classihed^i^^^^lB^ in accordance with security 
regulations of the War and NavyDepa^mei^s because certain chapters 
contain material which was at the date of printing. 

Other chapters may have had a lower classification or none. The reader 
is advised to consult the War and Navy agencies listed on the reverse 
of this page for the current classification of any material. 








Manuscript and illustrations for this volume were prepared 
for publication by the Summary Reports Group of the 
Columbia University Division of War Research under con¬ 
tract OEMsr-1131 with the Office of Scientific Research and 
Development. This volume was printed and bound by the 
Columbia University Press. 

Distribution of the Summary Technical Report of NDRC 
has been made by the War and Navy Departments. Inquiries 
concerning the availability and distribution of the Summary 
Technical Report volumes and microfilmed and other refer¬ 
ence material should be addressed to the War Department 
Library, Room lA-522, The Pentagon, Washington 25, D. C., 
or to the Office of Naval Research, Navy Department, Atten¬ 
tion : Reports and Documents Section, Washington 25, D. C. 

Copy No. 

239 


This volume, like the seventy others of the Summary Tech¬ 
nical Report of NDRC, has been written, edited, and printed 
under great pressure. Inevitably there are errors which have 
slipped past Division readers and proofreaders. There may 
be errors of fact not known at time of printing. The author 
has not been able to follow through his writing to the final 
page proof. 

Please report errors to: 

JOINT RESEARCH AND DEVELOPMENT BOARD 
PROGRAMS DIVISION (STR ERRATA) 

WASHINGTON 25, D. C. 

A master errata sheet will be compiled from these reports 
and sent to recipients of the volume. Your help will make 
this book more useful to other readers and will be of great 
value in preparing any revisions. 



SUMMARY TECHNICAL REPORT OF DIVISION 6, NDRC 


VOLUME 12 


DECLASSIFIED 
By authority Secretary of 


SEP 7 1960 

D ESI Ct -Al E^efonse memo 2 August 1960 

CONSTRUCTIONS'^”*’ 
CRYSTAL TRANSDUCERS 


LC REGULATION: BEFORE SERVICING 
OR REPRODUCING ANY PART OF THIS 
DOCUMENT, ALL CLASSIFICATION 
MARKINGS must BE' CANCELLglT 


OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT 

VAXNEVAR BUSH, DIRECTOR 

NATIONAL DEFENSE RESEARCH COMMITTEE 
JAMES B. CONANT, CHAIRMAN 

DIVISION 6 
JOHN T. TATE, CHIEF 


WASHINGTON, D. C., 1946 







NATIONAL DEFENSE RESEARCH COMMITTEE 


James B. Conant, Chairman 
Richard C. Tolman, Vice Chairjiian 
Roger Adams Army Re{)resentati^•e^ 

Frank B. Jewett Navy Representative- 

Karl T. Compton Commissioner of Patents-'* 

Irvin Stewart, Executive Secretary 


^Ar?? 2 y representatives in order of service: 

Maj. Gen. G. V. Strong Col. L. A. Denson 

Maj. Gen. R. C. Moore Col. P. R. Faymonville 

Maj. Gen. C. C. Wdlliams Brig. Gen. E. A. Regnier 

Brig. Gen. ^\^ ,4. ^Vood, Jr. Col. M. M. Irvine 

Col. E. A. Ronthean 


-A'GX'y representatives in order of sei'vice: 

Rear Adm. El. G. Bowen Rear Adin. J. A. Eurer 

Capl. Lybrand P. Smith Rear Adm. A. H. \'^an Kenren 

Gommodore H. Schade 
^Commissioners of Patents in order of service: 
CoiiAvay P. Coe Casper \V. Ooms 


NOTES ON THE ORGANIZATION OE NDRC 


The duties of the National Defense Research Committee 
were (1) to recommend to the Director of OSRD suitable 
projects and research programs on the instrumentalities 
of warfare, together with contract facilities for carrying 
out these projects and programs, and (2) to administer 
the technical and scientific work of the contracts. More 
specifically, NDRC functioned by initiating research pro¬ 
jects on requests from the Aimy or the Navy, or on re¬ 
quests from an allied government transmitted through 
the Liaison Office of OSRD, or on its own considered in¬ 
itiative as a result of the experience of its members. Pro¬ 
posals prepared by the Division, Panel, or Committee for 
research contracts for performance of the work involved 
in such projects were first reviewed by NDRC, and if ap¬ 
proved, recommended to the Director of OSRD. Upon 
approval of a proposal by the Director, a contract per¬ 
mitting maximum flexibility of scientific effort was ar¬ 
ranged. The business aspects of the contract, including 
such matters as materials, clearances, vouchers, patents, 
priorities, legal matters, and administi'ation of patent 
matters were handled by the Executive Secretary of OSRD. 

Originally NDRC administered its work through five 
divisions, each headed by one of the NDRC meml)ers. 
These tvere: 

Division .A — Armor and Ordnance 
Division B — Bombs, Fuels, Gases, & Chemical Problems 
Division C — Communication and Transportation 
Division D — Detection, Controls, and Instruments 
Division E — Patents and Inventions 


In a reorganization in the fall of 1942, twenty-three ad¬ 
ministrative divisions, panels, or committees w'ere created, 
each with a chief selected on the basis of his outstanding 
work in the particular field. The NDRC members then 
liecame a reviewing and advisory group to the Director 
of OSRD. The final organization Avas as follows: 

Division 1 — Ballistic Research 

Division 2 — Effects of Impact and Explosion 

Division 3 — Rocket Ordnance 

Division 4 — Ordnance Accessories 

Division 5 — New Missiles 

Division 6 — Sub-Surface Warfare 

Division 7 — Eire Control 

Division 8 — Explosives 

Division 9 — Chemistry 

Division 10 — Absorbents and Aerosols 

Division 11 — Chemical Engineering- 

Division 12 — Transportation 

Division 13 — Electrical Communication 

Division 14 —Radar 

Division 15 — Radio Coordination 

Division 16 — Optics and Camouflage 

Division 17 — Physics 

Division 18 — War Metallurgy 

Division 19 — Miscellaneous 

Applied Mathematics Panel 

Applied Psychology Panel 

Committee on ProjDagation 

T topical Deterioration Administrative Committee 



NDRC FOREWORD 


A s EVENTS of the years preceding 1940 re- 
^ vealed more and more clearly the serious¬ 
ness of the world situation, many scientists in 
this country came to realize the need of organ¬ 
izing scientific research for service in a national 
emergency. Recommendations which they made 
to the White House were given careful and sym¬ 
pathetic attention, and as a result the National 
Defense Research Committee [NDRC] was 
formed by Executive Order of the President in 
the summer of 1940. The members of NDRC, 
appointed by the President, were instructed to 
supplement the work of the Army and the Navy 
in the development of the instrumentalities of 
war. A year later, upon the establishment of the 
Office of Scientific Research and Development 
[OSRD], NDRC became one of its units. 

The Summary Technical Report of NDRC is 
a conscientious effort on the part of NDRC to 
summarize and evaluate its work and to present 
it in a useful and permanent form. It comprises 
some seventy volumes broken into groups cor¬ 
responding to the NDRC Divisions, Panels, and 
Committees. 

The Summary Technical Report of each Divi¬ 
sion, Panel, or Committee is an integral survey 
of the work of that group. The first volume of 
each group’s report contains a summary of the 
report, stating the problems presented and the 
philosophy of attacking them, and summarizing 
the results of the research, development, and 
training activities undertaken. Some volumes 
may be “state of the art” treatises covering 
subjects to which various research groups have 
contributed information. Others may contain 
descriptions of devices developed in the labora¬ 
tories. A master index of all these divisional, 
panel, and committee reports which together 
constitute the Summary Technical Report of 
NDRC is contained in a separate volume, which 
also includes the index of a microfilm record of 
pertinent technical laboratory reports and ref¬ 
erence material. 

Some of the NDRC-sponsored researches 
which had been declassified by the end of 1945 
were of sufficient popular interest that it was 
found desirable to report them in the form of 
monographs, such as the series on radar by 
Division 14 and the monograph on sampling 
inspection by the Applied Mathematics Panel. 
Since the material treated in them is not du- 


declassified 

By authority Secretary of 
plicated in the Summary Technical Report of 
NDRC, the monographs j^n impcmtant part 
of the story of these aspecrApf MDRU ©Search. 

In contrast to the information on radar, 
which is of wide^r^aifednteim8t2aAdtoS^lli9fl(§ 
which is released to the publier-fhe-research on 
subsurface warfarJ^5iIk^§My(M^s^MNHSiMS& 
of general interest to a more restricted group. 
As a consequence, the report of Division 6 is 
found almost entirely in its Summary Technical 
Report, which runs to over twenty volumes. 
The extent of the work of a Division cannot 
therefore be judged solely by the number of 
volumes devoted to it in the Summary Technical 
Report of NDRC: account must be taken of 
the monographs and available reports published 
elsewhere. 

Any great cooperative endeavor must stand 
or fall with the will and integrity of the men 
engaged in it. This fact held true for NDRC 
from its inception, and for Division 6 under the 
leadership of Dr. John T. Tate. To Dr. Tate and 
the men who worked with him—some as mem¬ 
bers of Division 6, some as representatives of 
the Division’s contractors—belongs the sincere 
gratitude of the Nation for a difficult and often 
dangerous job well done. Their efforts contrib¬ 
uted significantly to the outcome of our naval 
operations during the war and richly deserved 
the warm response they received from the Navy. 
In addition, their contributions to the knowl¬ 
edge of the ocean and to the art of oceano¬ 
graphic research will assuredly speed peacetime 
investigations in this field and bring rich bene¬ 
fits to all mankind. 

The Summary Technical Report of Division 6, 
prepared under the direction of the Division 
Chief and authorized by him for publication, 
not only presents the methods and results of 
widely varied research and development pro¬ 
grams but is essentially a record of the un¬ 
stinted loyal cooperation of able men linked in 
a common effort to contribute to the defense 
of their Nation. To them all we extend our 
deep appreciation. 

Vannevar Bush, Director 
Office of Scientific Research and Development 

J. B. CONANT, Chairman 
National Defe^ise Research Committee 


V 












FOREWORD 


A TRANSDUCER, as described in this volume, is 
a device capable of converting electrical 
energy into sound energy when used as a pro¬ 
jector, and capable of converting sound energy 
into electrical energy when used as a receiver. 
Although in an echo-ranging set the transducer 
serves this dual purpose, in many other applica¬ 
tions, the transducer is employed as a sound 
projector only or as a sound receiver only. In 
practically all of the many military applications 
of underwater sound, a transducer of one type 
or another is required. It was appropriate, there¬ 
fore, that adequate provision should be made 
for research upon the physical principles under¬ 
lying transducer design. 

It is possible to construct transducers employ¬ 
ing any one of a number of different physical 
principles. However, it is believed that the most 
satisfactory structures for sonar applications 
utilize either the magnetostriction effect ex¬ 
hibited by certain magnetic materials such as 
nickel and certain of its alloys, or the piezo¬ 
electric effect exhibited by certain crystals, 
such as quartz, Rochelle salt, and ammonium 
dihydrogen phosphate. 

The research undertaken by the Division was 
principally concerned with transducers of these 
two types. The results of studies pertinent to 
the design of magnetostriction transducers and 
materials are reported by Dr. Sabine of the 
Harvard University Underwater Sound Labora¬ 
tory in Volume 13 of this series. This volume 


reports the results of studies of transducer de¬ 
sign involving the piezoelectric effect. It should 
be recognized that the research of the Division 
summarized in these volumes was a continua¬ 
tion or expansion of previous work, and supple¬ 
mented other research in progress. 

Prior to the war both British and U. S. service 
laboratories had developed effective transducers 
for incorporation in echo-ranging sets and for 
use in other equipment. During the war period, 
the Navy continued its research both in the 
Naval Research Laboratory and through sup¬ 
port and sponsorship of research and develop¬ 
ment by industrial organizations. The total re¬ 
search and development effort was therefore 
very considerable and very substantial advance 
was made. 

The preparation of this report was under¬ 
taken by members of the U. S. Navy Electronics 
Laboratory (the Division’s San Diego Labora¬ 
tory, UCDWR, until March 1945), and the Di¬ 
vision is grateful to the authors and to the 
Navy for making possible the completion of this 
report after the San Diego Laboratory was 
transferred to Navy direction. The transducer 
research program received constant support 
from the Navy and splendid cooperation be¬ 
tween the various agencies engaged in research 
and development was secured for cordial inter¬ 
change of information. 

John T. Tate 
Chief, Division 6 



vii 









PREFACE 


T his volume seeks to give an account of the 
present state of transducer theory and of 
the art of transducer construction. Although 
a serious effort has been made by its authors 
to describe the work and thinking of other 
laboratories, it is inevitable that such a volume 
will be biased toward the home laboratory. 
Because they are better known to the present 
authors, the techniques practiced by and trans¬ 
ducers built by the University of California 
Division of War Research [UCDWR] are given 
more space than those of other laboratories. 

The general level of the book is variable. 
Some sections, such as those on construction 
techniques, can be read by anyone having only 
a slight technical background. Others require 
an extensive training in theoretical physics in 
order to appreciate their content. This is par¬ 
ticularly true of the basic discussion in Chap¬ 
ter 2. However, since the various portions are 
not interdependent, the book should be gen¬ 
erally useful to all engaged in the transducer 
art. 

This volume has many authors and the vari¬ 
ous chapters must be regarded as a contribution 
from the author or authors listed. The editor 
has refrained from altering them in any way 
and, as a result, some discontinuity of style 
may be noticed. As the final writing coincided 
in time with the conclusion of UCDWR’s activ¬ 
ities and with the resulting dispersion of per¬ 
sonnel, it has not been possible to check the 
interchapter references as closely as would be 
desired. The mathematical sections have not 
had the advantage of being smoothed by class¬ 
room use and may present undue difficulty to 
the reader. 

Shortly after the beginning of UCDWR it 
became clear that the work in both fundamental 
research and engineering development would 
require special transducers. It seemed more 
expedient to establish a group, the Transducer 
Laboratory, within its organization than to at¬ 
tempt to have such units built outside. It was 
fortunate, indeed, that the publication of the 
excellent book by W. P. Mason, Electromechmi- 
ical Transducers and Wave Filters, came when 
it did, for it has served as a textbook for many 


of the UCDWR scientists who were learning 
the art. Some early work was done, prior to the 
establishment of the Transducer Laboratory, 
by D. K. Froman, A. M. Thorndike, C. H. Kean, 
and G. E. Duval on the properties of X-cut 
Rochelle salt crystals. This study was useful in 
that it confirmed and extended the known data 
on this material. 

The need for transducers early in World 
War II was so great that it was not possible to 
devote the time or manpower to the desired 
fundamental studies of piezoelectricity and elas¬ 
ticity which were needed to give a firm basis 
for transducer design procedures. UCDWR was 
forced for nearly two years to build transducers 
by cut-and-try methods and profit as best it 
could by experience gained with them. 

Not until late 1943 was it possible to make 
a reassignment in personnel at UCDWR so that 
a research group could be set up within the 
Transducer Laboratory. The members of this 
group operated under the general assignment 
to learn whatever they could about transducers. 
Some of this knowledge came from visits and 
conferences with other transducer groups 
throughout the country, the remainder from 
theoretical and experimental research in San 
Diego. This volume is a compilation of such 
accumulated information. 

G. D. Camp, who was in charge of the re¬ 
search group just mentioned, was responsible 
for the arrangement of this volume and in gen¬ 
eral guided the work of the group which ob¬ 
tained much of the original data given herein. 
T. F. Burke, Jr., one of the original members of 
the Transducer Laboratory, originated many of 
the design procedures which have been used by 
UCDWR. B. G. Eaton, F. M. Uber, and F. X. 
Byrnes have all made plentiful contributions; 
Eaton has done much work'on directivity pat¬ 
terns and has made laboratory studies on many 
phases of the transducer art. Uber, in the short 
time he was with UCDWR, made many investi¬ 
gations in methods of attaching crystals to 
various surfaces and devised production con¬ 
trols for cemented joints. He is solely respon¬ 
sible for the introduction of the Cycle-Welding 
technique for the attachment of crystals to rub- 


IX 


X 


PREFACE 


ber. Byrnes has been engaged in the design of 
several of the UCDWR transducers and in 
methods of matching electronic circuits to them. 

The editor desires to acknowledge the im¬ 
portance of the roles played by many others 
who, while not authors of this volume, have 
through their work very largely made it possi¬ 
ble. Special mention is made of G. A. Argabrite 
who relieved the undersigned as head of the 
Transducer Laboratory in the early months of 
its growth and remained its leader throughout 
the war. There is scarcely a phase of the work 
of this group which has not profited by his 
active enthusiasm. D. C. Kalbfell, as a member 
of the research group, was concerned signifi¬ 
cantly with the electronic equipment associated 
with transducers. He is particularly to be men¬ 
tioned for his studies of impedance bridges. 
R. Bellman did much of the original theoretical 
work in Chapter 3. H. W. Hunter did valuable 
work on all phases of transducer construction 
being particularly concerned with their rugged¬ 
ness or resistance to shock. K. M. Burton, as 
foreman of the transducer shop, by his pains¬ 
taking efforts made transducer construction 
more reliable. C. E. Green, F. L. Paul, and Miss 
M. C. MacKenzie conducted many laboratory 
experiments for the research group. The many 
mathematical and numerical calculations in¬ 
volved in the development of some of the con¬ 
clusions presented in this volume were done by 
a group of mathematicians consisting of Mrs. 
A. J. Keith, Mrs. 0. W. Wilt and Mrs. F. C. 
Herreshoff. 

Most particular mention must be made of the 
Calibration Laboratory of UCDWR whose task 
it was to assay the performance of all units 


designed by the Transducer Laboratory. C. J. 
Burbank headed this group and together with 
J. H. Martin, head of the Sweetwater Calibra¬ 
tion Station, and the latter’s predecessor, D. H. 
Ransom, offered on the basis of their calibra¬ 
tions many useful suggestions to the transducer 
designers. Without the able help of this group 
little could have been accomplished by the 
Transducer Laboratory. 

As the manuscript of this volume neared 
completion, D. J. Evans, assisted by C. A. Young 
and L. A. Cartwright undertook the mechanical 
task of assembling illustrations, paging, and 
numbering, which constitute the final, most 
tiresome labors. The art work for illustrations 
in Chapter 8 was done by S. F. Simonet. Also 
contributing to Chapter 8 were Theron Lam¬ 
bert, G. W. Banks, and V. G. McKenny. 

Finally, the excellent cooperative spirit with 
which members of UCDWR were always re¬ 
ceived by the Bell Telephone Laboratories, the 
Brush Development Company, the Submarine 
Signal Company, and the Naval Research Lab¬ 
oratory has been very helpful throughout World 
War 11. To mention a few of the gentlemen in 
these companies who have been particularly 
stimulating one must include A. C. Keller, W. H. 
Martin, and W. P. Mason of the Bell Telephone 
Laboratories; A. L. Williams, W. R. Burwell, 
Frank Massa, and Harry Shaper of the Brush 
Development Company; H. J. W. Fay and I. C. 
Clement of the Submarine Signal Company; 
and H. C. Hayes and E. B. Stephenson of the 
Naval Research Laboratory. 

F. N. D. Kurie 
E ditor 




CONTENTS 


CHAPTER PAGE 

1 General Survey by G/e/z D. Camp . 1 

2 The Phenomenological Theories of Linear Dissipative Elec¬ 
trics, Dielectrics, and Piezoelectrics by G/ezz D. Czizzi/; . . . 30 

3 Properties of the Component Parts of Crystal Transducers by 

Richard Bellman, T. Finley Burke, Glen D. Camp, Bourne G. 
Eaton, ■a.nd Fred M. Uber .73 

4 Properties of Assembled Crystal Transducers by T. Finley 

Burke, Glen D. Camp, and Bourne G. Eaton .127 

5 Electronic Systems and Matching Networks by Francis X. 

Byrnes .211 

6 Design Procedures by T. Finley Burke .230 

7 Design Adjustment by T. Ehz/^y 5?/r/zc.255 

8 Construction Techniques and Equipment by Tzcd M. f/5er . 267 

9 Research Techniques and Apparatus by T. Finley Burke, 

Francis X. Byrnes, and Bourne G. Eaton .358 

Glossary.381 

Bibliography.383 

Contract Numbers.387 

Index.389 
















Chapter 1 

GENERAL SURVEY 

By Glen D. Camp 


II INTRODUCTION 

T his volume represents an attempt at a 
unified account of present knowledge rele¬ 
vant to the design and construction of crystal 
transducers; some account of the work of other 
laboratories has been given, but this is by no 
means complete. 

The basic purpose of underwater transducers 
is to generate and receive sonic signals in water. 
These signals may be used for the detection of 
submerged objects, the control of devices, inter¬ 
ference with the operation of enemy devices, 
and underwater communication. These applica¬ 
tions are discussed here only to the minimum 
extent necessary to understand the various 
transducer characteristics needed. 

Underwater transducers are functionally 
equivalent to radio and radar antennae. Acting 
as a transmitter, an electric signal sets a part 
of the transducer, which is often called the 
“motor,’’ in motion and thus produces an out¬ 
going signal. As a receiver, the process is re¬ 
versed : an incident sonic signal sets the motor 
in motion, producing an electric signal at the 
terminals. Both transducer and antenna are 
inert systems, requiring an external driver or 
receiver amplifier; both are coupled to a field, 
the former elastic and the latter electromag¬ 
netic ; and in both, the field exists and is propa¬ 
gated in a medium, one being an elastic fluid, 
water, and the other, empty space or the 
“ether.” 

In crystal transducers the fundamental phys¬ 
ical phenomenon, whereby energy is converted 
from one form to another, is piezoelectricity. 
Historically, this phenomenon is divided into 
the direct and inverse effects. The direct effect 
is the production of an electric field in a crystal 
when deformed, and was discovered by the 
Curie brothers in 1880; it is the effect that 
makes a receiver possible. The inverse effect 
is the distortion of a crystal when exposed to 
an electric field, and makes a transmitter pos¬ 


sible. These effects are not independent but, 
like all reversible processes, are inextricably 
connected by a reciprocity principle. In fact, 
the next year after the Curies discovered the 
direct effect, Lippman predicted the existence 
of the inverse effect, on the basis of their dis¬ 
covery and the thermodynamic laws of reversi¬ 
ble processes. His prediction was verified by 
the Curies that same year, although this dis¬ 
covery might have been delayed many years if 
the prediction had not suggested what to look 
for. 

Piezoelectricity, in common with magnetism, 
dielectrics, etc., has been studied from two quite 
independent points of view. The first is molec- 
idar piezoelectricity, which is concerned with 
the fundamental interactions which cause the 
phenomenon. The second is phenomenological 
piezoelectricity, in which one ignores the fun¬ 
damental cause completely but seeks to obtain 
a correct description of the behavior of macro¬ 
scopic crystals. 

In this volume, we are completely uncon¬ 
cerned with the molecular aspect of piezoelec¬ 
tricity. A macroscopic crystal is regarded as 
a given thing, having certain complicated prop¬ 
erties susceptible to gross experimental study. 
Our only concern with piezoelectricity is to 
understand these macroscopic properties suffi¬ 
ciently well to enable us to use crystals for a 
specific purpose.-'^ While piezoelectricity is the 
basic phenomenon which permits a crystal 
transducer to function, we are also deeply con¬ 
cerned with other branches of physics which 
become involved when one attempts to put the 
basic phenomenon to a useful purpose. In fact, 
in few branches of applied physics is the theory 

This remark refers solely to the contents of this 
volume. Research in molecular piezoelectricity at Brush 
Development Company and Bell Telephone Laboratories 
led, in the midst of World War II, to the almost com¬ 
plete displacement of Rochelle salt by ammonium dihy¬ 
drogen phosphate, the latter having many advantages 
over the former. It is possible that continuation of this 
work might lead to crystals still better suited to the 
construction of transducers. 


1 



2 


GENERAL SURVEY 


more complicated, the phenomena more diverse, 
the experimental procedures more difficult, or 
the techniques more critical. Nearly all prewar 
research was done on highly idealized cases 
with the emphasis on pure piezoelectricity. 
Quite properly, for the purposes of that re¬ 
search, every effort was made to eliminate just 
those complicated interactions which play an 
essential role in the performance of a practical 
transducer. Here we must take an entirely dif¬ 
ferent attitude: we must recognize that this 
idealized approach is only an important first 
step and we should attempt, by a closely re¬ 
lated program of theoretical and experimental 
research and field tests, to learn what factors 
interfere with the minute motions of crystals 
and how to control these motions sufficiently to 
put them to practical use. 

For our present purpose, a crystal is an ob¬ 
ject such that, if a properly oriented piece is 
cut from the mother bar, equipped with suitable 
electrodes and properly mounted and protected, 
it will serve to generate or receive a sonic 
signal. As will develop later, rectangular plates 
cut at certain angles from mother bars of 
Rochelle salt [RS] or ammonium dihydrogen 
phosphate [ADP], and designated as 45° X-cut 
and 45° Y-cut RS and 45° Z-cut ADP (see 
Figure 1), are the only types of cut crystals 
that have so far found extensive practical ap¬ 
plication in underwater transducers in the 
United States. Quartz has been effectively used 
in England, but only because an adequate 
supply of the above synthetically grown crystals 
was not available there. For reasons to be dis¬ 
cussed later, the use of 45° X-cut RS is now 
regarded as a regrettable expedient to be 
tolerated only in very special and rare circum¬ 
stances. The use of 45° Y-cut RS has greatly 
declined, but is still of sufficient importance to 
warrant inclusion. This volume therefore deals 
with 45° Y-cut RS and 45° Z-cut ADP almost 
exclusively, touching only briefly on 45° X-cut 
RS for small hydrophones on long cables where 
a preamplifier cannot be used. 

In dealing with these rectangular plates of 
RS or ADP we are here concerned (1) with 
tests for verifying that they are properly ori¬ 
ented and that they satisfy certain other stand¬ 
ards; (2) with the best sizes and shapes for 


a particular application; (3) with the best 
ways of mounting and protecting them; (4) 
with their coupling to the water; (5) with the 
effects produced, how these depend upon fre¬ 
quency, and the response to pulse excitation. 
We are concerned also with the improvement 
and standardization of the numerous tech¬ 
niques, both experimental and constructional, 
which are involved, and with the design of 
suitable electronic drivers for transmitters and 
suitable amplifiers for receivers (the electric 
circuit characteristics of crystal transducers 
are sufficiently specific and critical to warrant 
special study of this electronic problem). Above 
all, we are concerned with knowing how, as 
far as is at present possible, to produce a 
transducer which will serve a specific purpose 
in a specific device and which will be practical 
to manufacture in quantity. 

Crystal transducers are devices with a 
sharply limited domain of useful applicability. 
Their performance is not sufficiently flexible 
to allow one intended for one purpose to serve 
well for another. Only by a thorough knowledge 
of the factors which influence their behavior, 
together with a clear understanding of the 
effects which it is desired to achieve, can one 
hope to produce a transducer which will be 
satisfactory for a given service. In fact, the 
first list of desirable characteristics almost in¬ 
variably contains features which are contra¬ 
dictory. Only after careful consideration of all 
aspects of the problem, in the light of all avail¬ 
able knowledge, is it usually possible to reach 
a workable compromise between desired and 
realizable characteristics. 

While this circumstance arises, to a consid¬ 
erable extent, from inadequate knowledge and 
techniques, a part of it is intrinsic to the 
crystal itself and can be alleviated only by an 
entirely different line of research, namely, find¬ 
ing a crystal with more suitable properties. 
As previously stated, this volume makes no 
attempt to follow this attack. The ideal toward 
which we work is therefore to remove all un¬ 
satisfactory features which are extrinsic to 
the crystal so that the final result is limited 
only by the properties of the best available 
crystals, always taking account of the practica¬ 
bility of manufacture. Much progress has been 



INTRODUCTION 


3 



X 



Figure 1, Orientation of rectangular plates of 45° X-cut and Y-cut RS (top) and 45° Z-cut ADP 
(bottom), in the mother bars. 


















4 


GENERAL SURVEY 


made in this direction, but there is still much 
to be done. 

It will pay us to pause here to summarize the 
foregoing basic considerations and to define as 
sharply as possible the purpose, scope, and 
limitations of this volume. 

1. Underwater crystal transducers are de¬ 
vices which use the piezoelectric effect to con¬ 
vert electric and sonic signals reciprocally. 
They have a variety of applications, discussed 
in this volume only to the minimum extent nec¬ 
essary to understand the origin of specifica¬ 
tions. 

2. Piezoelectricity has molecular and phe¬ 
nomenological aspects. Only the latter, sufficient 
for the purpose of this volume, is discussed 
here; however, further study of the molecular 
aspects, which has already made a great con¬ 
tribution in developing ADP, might make fur¬ 
ther contributions. Only rectangular plates of 
45° Y-cut RS and 45° Z-cut ADP are exten¬ 
sively treated here, these sufficing for the 
great majority of underwater crystal trans¬ 
ducers at present. 

3. Although piezoelectricity is the basic, nec¬ 
essary phenomenon, other branches of physics 
play essential roles in the behavior of crystal 
transducers. These are complicated and critical. 

4. The characteristics of crystal transducers 
are not very flexible, and they must therefore 
be designed and built to serve a specific purpose 
and will usually not serve well in a service for 
which not intended. 

5. The overall purpose of this book is to pre¬ 
sent, in as thorough a form as possible, a com¬ 
plete account of all aspects of present knowl¬ 
edge of the problem of producing underwater 
crystal transducers which will serve a specified 
purpose, the ultimate goal being a finished 
product which is practical to manufacture in 
quantity, and which will do a specified job in 
service. 


TYPICAL UNITS 

Underwater crystal transducers are built in 
a wide variety of external shapes, sizes, and 
internal constructions. Some idea of external 


shapes and sizes will be gained from Figure 2. 
A brief identification of the units follows. 

1. An early Bell Telephone Laboratories unit, 
built for a particular research application, 
which operates in the supersonic frequency 
band and may be used either as a receiver or as 
a transmitter. 

2. A Brush Development Company unit, used 
for listening in the sonic frequencies. Its 
unusual shape results from its containing a 
parabolic reflector which focuses the sound on 
a small crystal assembly. 

3. The University of California Division of 
War Research [UCDWR] CQ8Z, the transmit¬ 
ter-receiver of QUA sonar, a device which was 
manufactured in considerable quantity (con¬ 
sidering its large size) and assisted U. S. sub¬ 
marines in evading mines. It went through sev¬ 
eral modifications and the model shown, the 
latest, has many interesting features. Perhaps 
the most unusual is the high degree of acoustic 
isolation between the transmitting and receiv¬ 
ing motors, which makes the “crosstalk” level 
very low. (See Chapter 6.) 

4. The UCDWR JB has a spherical shape and 
general external appearance very similar to 
several Submarine Signal Company transduc¬ 
ers, but the internal construction is quite dif¬ 
ferent. 

5. The UCDWR GD case was used for a 
number of different units all practically identi¬ 
cal in external appearance, but differing widely 
in internal construction. An example of these 
is the GD-16 (see Figure 34). 

6. The UCDWR BE is an example of an un¬ 
usual shape made necessary by the application 
to which the transducer was put. Sound is radi¬ 
ated through the curved part of the case. 

7. The UCDWR BG, later and improved 
model of the UCDWR BE. This is shown in 
detail in Figure 35. 

8. The UCDWR KC is an excellent trans¬ 
ducer which was produced in considerable quan¬ 
tity during World War II. The case is made of 
very thin metal in a streamlined shape, except 
for top and bottom members which are made 
heavier for strength and isolation. Despite the 
thinness of the wall, it is amply rugged for its 
purpose. 

9. This Submarine Signal Company unit con- 



TYPICAL UNITS 


5 



FIGURE 2. Typical crystal transducers, illustrating comparative sizes and shapes. (See text for 
explanation.) 


_ 










6 


GENERAL SURVEY 



Figure 3. The crystal array of a typical trans¬ 
ducer. The individual crystal blocks are sepa¬ 
rated by isolating material and bound together 
as a single unit in a frame. All the crystal 
blocks are not the same size so as to regulate 
the shape of the beam pattern. This unit is in¬ 
tended for operation at supersonic frequencies. 
(Bell Telephone Laboratories D172735 modified 
QB transducer.) 



Figure 4. Another view of the same unit shown 
in Figure 3. This is the completed “motor” or 
vibrating element of the transducer. The crystals 
are on the right side of a metal plate which has 
been cut up into a number of resonating bars 
whose purpose is to modify the resonant fre¬ 
quency of the unit. The crystal array is attached 
to this backing plate by means of a cement or 
glue. (Bell Telephone Laboratories D172735 
modified QB ti’ansducer.) 




Figure 5. The assembly of Figure 7 mounted 
in its case not yet entirely closed. The right side 
of the case is made of qc rubber through which 
supersonic sound passes freely. (Bell Telephone 
Laboratories D172735 modified QB transducer.) 



I ^ V a. n 


Figure 6 . The motor of a transducer showing 
octagonal array of crystals cemented to the 
backing plate, the outermost crystals being twice 
as large as those in the central section to control 
the shape of the directivity pattern. The vertical 
lines, visible under some of the larger crystals, 
are grooves in ceramic wafers which are 
cemented between the crystals and the back¬ 
ing plate for voltage insulation. 



























TYPICAL UNITS 


7 


tains an unusual motor, consisting of a number 
of Benioff blocks, the theory of which is dis¬ 
cussed in Chapter 3. 

10. The UCDWR CY4 is a simple stack 
motor, sealed in an olive can. It was also pro¬ 
duced in large quantities during World War II. 

11. The UCDWR EP is a “window-coupled” 
unit, the crystals being Cycle-Welded on the 
inside to the rubber window. More details on 
an earlier model of this transducer are shown 
in Figures 36 and 37. 

12. The UCDWR CD or CJ is a stack motor 
in a rubber sleeve, strengthened by a perforated 
tube. These were used in Navy model OAS and 
OAU practice targets for training purposes. 

13. The UCDWR probe microphone or acous¬ 
tic ammeter is a late model of a device for 



Figure 7. The motor of Figure 4 is shown here 
attached to a baffle which also carries a coil as¬ 
sembly. This assembled unit is ready to be put in 
a case which will subsequently be filled with 
castor oil to provide a path for sound from the 
motor through the case to the surrounding water. 
The purpose of the baffle, which is a good ab¬ 
sorber of sound at supersonic frequencies, is to 
suppress backward radiation from the resonating 
bars. The coils modify the electrical character¬ 
istics of the unit to match it to its driving 
amplifier. (Bell Telephone Laboratories D172735 
modified QB transducer.) 


studying the motion of a surface. The active 
element is a minute crystal at the tip, enclosed 
in a circle in the picture. The cable, which con¬ 
tains a guard shield, goes to a cathode-follower 
amplifier. 

Several of these units will be studied in some 
detail in later parts of this volume. For the 
present, it may be useful to give a general idea 
of the internal construction of typical trans¬ 
ducers. The views of assembled and disassem¬ 
bled units shown in Figures 3 to 40 should 
be sufficiently self-explanatory for the present. 

The purpose of this section is to make evident 
the broad range of shape, size, and other prop¬ 
erties required to produce transducers which 
will satisfactorily perform the wide variety of 
functions demanded in service. 



Figure 8. The completely assembled trans¬ 
ducer. The oc rubber sound window is still on 
the right. The unit has now been filled with 
castor oil and is ready for use. The flange at the 
top will be used to attach it to a shaft protruding 
through the bottom of a ship. The lead which 
connects the crystal motor to the rest of the 
electronic system will enter the ship through 
this shaft. (Bell Telephone Laboratories D172735 
modified QB transducer.) 






8 


GENERAL SURVEY 



Figure 9. Another view of the transducer motor 
shown in Figure 6. 


Figure 11. The baffle for the transducer ^ of 
Figure 6. This is mounted immediately behind 
the resonating bars shown in Figure 10 to ab¬ 
sorb sound radiated in this undesired direction. 
It consists of closely spaced sheets of fine mesh 
metal gauze in an expanded metal frame. The 
sound absorption mechanism is viscous friction 
of castor oil, moving in the many narrow pas¬ 
sages formed by the metal gauze. (Bell Tele¬ 
phone Laboratories D171932 MCC transducer.) 






Figure 10. The reverse side of the backing plate 
of the motor in Figure 6. This plate has been cut 
so that each crystal block is backed by its own 
resonating bar. The cuts are intended to allow 
each bar to vibrate independently of the others. 
(Bell Telephone Laboratories D171932 MCC 
transducer.) 


Figure 12. The completed transducer shown in 
Figure 11. The darker section is the qc rubber 
sound window through which sound is radiated. 
The cables and mounting flange may be seen at 
the top of the case. (Bell Telephone Laboratories 
1)171932 MCC transducer.) 
























TYPICAL UNITS 


9 


t 



RUBBER SHELL 


GLAND' 


RUBBER SHELL 


LEAD WEIGHT 


CRYSTAL 

MOTOR 


METAL LINER 


CABLE 


Figure 13. Disassembled nondirectional hydrophone. The crystal motor shown in Figure 15 is mounted 
inside a metal liner through holes in which the sound reaches the active ends of the crystals. This liner 
has a lead weight attached to keep it upright. The whole assembly fits in a two-piece rubber shell, the 
lower part of which is filled with castor oil. (Brush Development Company AX83 hydrophone.) 



Figure 14. The completely assembled Brush Development Company AX83 hydrophone. It is supported 
by its cable and hangs in the water with its long axis vertical. 














10 


GENERAL SURVEY 


PRESSURE 

RELEASE 

MATERIAL 



Figure 15. The motor of a nondirectional 
hydrophone designed for operation in the sonic 
frequency band. Two quite large crystals have 
their two active ends exposed but their active 
sides are covered by pressure release material. 
(Brush Development Company AX83 hydro¬ 
phone.) 



RUBBER SHELL 


CRYSTAL MOTORS 


MAIN SUPPORTING 
TUBE 






Figure 16. This transducer has several vertical strip motors immersed in castor oil in the space between 
the metal tube and the rubber shell. Separate leads are brought out from each motor. (Brush Develop¬ 
ment Company AX89 transducer.) 























TYPICAL UNITS 


11 





Figure 17. Another view of the Brush De¬ 
velopment Company AX89 transducer with the 
crystal motors in place. The crystal blocks in 
these motors are thicker at the upper and 
lower ends to control the shape of the direc¬ 
tivity pattern. 



Figure 18. The completely assembled Brush 
Development Company AX89 transducer. The 
rubber shell is sealed to the main metallic 
structure by clamping bands. The slight bulge 
in the rubber is caused by the pressure of the 
castor oil. This unit was designed to be used 
as a transmitter in the supersonic frequency 
band. 



Figure 19. A transducer similar to the one shown in the preceding figures. Here the crystals are 
nested in pressure-release material and are mounted in a manner similar to that shown in Figure 17. 
(Brush Development Company AX104 transducer.) 
































12 


GENERAL SURVEY 



Figure 20. The partly assembled motor of the Brush Development Company AX124 transducer. The 
crystal array is built in two sections, with pressure-release material separating and surrounding each 
block of crystals, and then cemented to a glass backing plate. 


CRYSTAL MOTOR 



Figure 21. The motor of Figure 20 shown in its case. The rubber sound window will be attached to 
this case by means of the clamping band and the unit will then be filled with castor oil. This transducer 
was designed as a transmitter of sound in the supersonic frequency band. (Brush Development Com¬ 
pany AX124 transducer.) 








TYPICAL UNITS 


13 



Figure 22. A large transducer designed for 
transmission at supersonic frequencies. The coni¬ 
cal assembly of the individual motors will give a 
slight uptilt to the beam. One of the separate 
motors may be seen in the foreground. Both 
the crystals and their backing bars are 
cemented together in a shell of pressure-release 
material. This unit is similar in construction 
and purpose to those shown in Figures 17 and 
19. (Brush Development Company AX132 
transducer.) 



Figure 23. The completely assembled Brush 
Development Company AX132 transducer. 


CRYSTAL MOTOR 



Figure 24. A disassembled hydrophone. The small block of crystals may be seen mounted in the middle 
of the main casting. (Brush Development Company AX133-5 transducer.) 















14 


GENERAL SURVEY 


DIAPHRAGMS 



CRYSTAL MOTOR 


Figure 25. Parts of a small calibration hydro¬ 
phone. The stack of three tiny crystals is 
mounted in the eye of the brass housing. The 
two copper diaphragms are cemented over this 
eye and the whole structure is covered first by 
copper and then by chromium plating. (Brush 
Development Company Cll hydrophone.) 



Figure 26. Three calibration hydrophones. 
From left to right, each is designed for pro¬ 
gressively lower frequency. The one at the 
extreme left is shown disassembled in Figure 
25. (Brush Development Company Cll, CIO, 
and C7 hydrophones.) 



Figure 27. Another hydrophone of the same 
general construction as shown in Figures 13 
to 15. (Brush Development Company C22-A2 
hydrophone.) 



Figure 28. A small hydrophone. This unit is 
interesting because the crystals are mounted 
“on their side.” (Brush Development Company 
C45 transducer.) 












TYPICAL UNITS 


15 




Figure 29 Figure 30 

Crystal array of NRL transducer is shown in Figure 29 with larger crystals around the edge to shape 
directivity pattern. Each block of crystals is cemented to a square backing plate carrying a cylindrical 
resonator rubber mounted in a small cylindrical case so that the resonator is surrounded on all sides by 
pressure-release material (air). For repair, each unit can be removed without disturbing the others. (See 
also Figure 13 of Chapter 3.) Figure 30 shows the back of the motor with the cups around each resonator. 
Unsoldering two leads and removing four screws releases any unit. (NRL.) 



Figure 31. Right, bakelite disk to which the separate units are attached, including left, part of a rubber 
sandwich used to seal the portion of the transducer containing castor oil from the remainder. (NRL.) 












16 


GENERAL SURVEY 



Figure 32. Completely assembled motor mounted on rubber sandwich ready for insertion in case. Tight¬ 
ening the various bolts compresses the rubber so that the section of case containing the crystals may be 
oil-filled. (NRL.) 



IMPEDANCE MATCHED 
OR "rho-c" rubber 

WINDOW 


ISOLATION 

MEMBER 


CYLINDRICAL 
TRANSMITTER 
80 DEGREE 


LOBE SUPPRESSED 
RECEIVER 


HALF-CYLINDER OF 
STEEL FOR STRENGTH 


Figure 33. Transducer with two separate motors used in sonar system which radiates continuously. The 
receiver (upper motor) is flat, producing a sharp directivity pattern (Figure 43), whereas the trans¬ 
mitter (lower motor) is cylindrical, producing a broad pattern (Figure 44). A baffle between the two 
minimizes the sound fed from the transmitter to the I’eceiver. (Assembled transducer, item 3 Figure 2 ) 
(UCDWR CJJ78256.) 




































TYPICAL UNITS 


17 



Figure 34. A high-frequency transducer having a motor in which the crystals are mounted “on their 
side” on a steel porcelain-covered backing plate. This motor is mounted in the case by nesting it in 
pressure-release material. The rubber window bonded into a metal frame is shown at the right. (UCDWR 
GD16 transducer.) 












18 


GENERAL SURVEY 



CRYSTAL MOTORS 


RUBBER CONNECTOR 


Figure 35. Extreme design occasioned by difficult specifications. Vibrating faces of the two motors are 
narrow in a horizontal plane and wide in a vertical plane. Motors are mounted in two thin metal cases 
connected by belt of rubber. The metal is sufficiently thin to transmit sound without appreciable attenua¬ 
tion. The result is a transmitter whose pattern is nearly circular in the horizontal plane and comparatively 
narrow in the vertical plane. (UCDWR BG2 transducer.) 



Figure 36. Small high-frequency transducer used in a hand-held echo-ranging device. Crystals are 
Cycle-Welded to rubber sheet in direct contact with the water. No backing plate. Since the crystals are 
coupled very directly through the thin rubber sheet with the water, this transducer is air-filled, not oil- 
filled. (UCDWR EP transducer.) Item 11, Figure 2 shows more advanced version of unit. 











TYPICAL UNITS 


19 





Figure 37. A step in the assembly of the UCDWR EP Transducer. The crystal array has been built up 
and is shown in its clamp at the left. The rubber diaphragm in an aluminum frame has been prepared 
for Cycle-Welding. The polarizing marks on the crystal in the array are to guide the assembler so that 
the crystals will be properly oriented to vibrate in phase. 












20 


GENERAL SURVEY 



Figure 38. A ti-ansducer designed to have a uniform directivity pattern in the horizontal plane and a 
sharp pattern in the vertical plane. The crystal motor consists of a spiral “stack” cemented to pressure- 
release material except on the active ends. The motor is mounted inside a perforated shell which is 
covered by a rubber sleeve and filled with castor oil. (UCDWR CD transducer.) This is item 12 of 
Figure 2. 


















TYPICAL UNITS 


21 



Figure 39. This transducer was required to meet very stringent specifications. Not only was its shape 
and size specified, but it was necessary that it withstand severe shock tests and that its frequency band 
be extremely large. For this latter reason, two separate motors have been made by Cycle-Welding 
crystals to a length of pressure-release material which is then wrapped around the central arbor. This 
unit, on being covered by the rubber sleeve, is castor-oil-filled. (UCDWR Type CCUIOZ.) 














22 


GENERAL SURVEY 



Figure 40. The interior of another transducer 
designed to withstand extreme shock. Rows of 
crystals are Cycle-Welded to a rubber sleeve 
which has strengthening metal inserts in it. 
The unit is air-filled. (UCDWR CCU6Z trans¬ 
ducer.) 

' 3 CALIBRATION DATA 

In evaluating the performance of completed 
crystal transducers, certain standardized, rou¬ 
tine calibration tests are in common use. There 
are many technical questions involved in the 
selection of apparatus and procedures for these 
tests and in determining their accuracy, which 
are not considered in this volume.^- ^ 

The purpose of this section is merely to 
enumerate these tests, as an indication of the 
type of information which may readily be ob¬ 
tained from a calibration laboratory. 

In the performance of calibration tests, the 
transducer is regarded as a “black box,” elas¬ 
tically coupled to the water and electrically ac¬ 
cessible through two or more leads. While it is, 
therefore, not always simple to interpret the 
results in terms of internal properties, the re¬ 
sults of these tests are nevertheless of the ut¬ 
most importance. 

' " ^ Ideal Medium 

Calibration tests should be made so as to 
depend as little as possible on the properties 


of the medium and its surroundings. For ex¬ 
ample, every precaution should be taken to 
eliminate effects arising from surface and bot¬ 
tom reflections, bubbles, and obstacles. The 
ideal is to obtain the characteristics of the unit 
in an infinite homogenous medium which is 
completely described, for our purposes, by its 
phase velocity and density. The effects produced 
by a transducer in an actual medium then con¬ 
stitute a transmission problem, and are not 
considered in this voliime,-"^ 

^ ^ ^ Directivity Patterns 

One important property of a transducer, 
when acting as a transmitter, is the manner in 
which transmitted energy is distributed in di¬ 
rection; or, when acting as a receiver, the de¬ 
pendence of its sensitivity on the direction of 
the incident radiation. 

The pressure produced by the unit acting as 
a transmitter, at sufficiently great distances 
(i.e., in the radiation field), can be shown to be 
given by T ie,<t))/r, in which 6, and r refer 
to a polar coordinate system with origin inside 
the unit. The function T, after multiplication 
by a normalizing factor to make it 1 in the di¬ 
rection of some suitably chosen axis, is called 
the pressure directivity pattern and may be a 
complex number; its absolute square is the in¬ 
tensity directivity pattern. 

If a plane wave of fixed amplitude is incident 
upon the unit from various directions, the po¬ 
tential difference developed across the terminals 
is and this function, after normaliza¬ 

tion, is called the pressure directivity pattern 
of the unit as a receiver; its absolute square 
is the intensity pattern. It is shown in Chap¬ 
ter 4, on the basis of very general assumptions, 
that the normalized T and R functions are 
identical; that is, the intensity-directivity pat¬ 
tern of a transducer should be the same whether 
it acts as a transmitter or a receiver. This has 
been experimentally verified repeatedly, so that 
now it is customary to measure either \T\ or \R\, 
whichever is more convenient. 

The directivity pattern of a transducer is 
usually plotted on a decibel scale, so that the 
quantity given is 10 log \T\~ or 10 log 
usually adjusted to 0 db in the direction of the 






CALIBRATION DATA 


23 




leo 

Figure 43. Directivity pattern of the receiver 
section of the unit shown in Figure 33, at 42 
kc. The small side lobes are particularly to be 
noted. Except for two bulges on the main lobe 
at about —25 db, there are no side lobes 
greater than —30 db. Test distance, 20 feet; 
depth, 9 feet. (UCDWR CJJ78256 transducer.) 


Figure 42. The horizontal directivity pattern 
of the unit shown in Figure 41. The maximum 
variation from uniformity is about ±3 db, 
which was not important for its application. 
(UCDWR CCU8Z transducer.) 


<60 

Figure 44. Directivity pattern of the trans¬ 
mitter section of the transducer shown in 
Figure 33. The application of curving the crys¬ 
tal array to get a broad beam is illustrated 
here. The typical diffraction pattern around 
the back portion is characteristic of units of 
this type. 


180 

Figure 41. Directivity pattern of the top 
motor of a transducer, similar to that shown 
in Figure 39, in a plane through the vertical 
axis. The radiation upward and downward is 
only about 15 db below maximum. Frequency 
60 kc; test distance, 20 feet; depth 9 feet. 
(UCDWR CCU8Z transducer.) 
































































24 


GENERAL SURVEY 




100 

Figure 45. The directivity pattern of a unit 
whose motor is long and narrow, taken in a 
plane containing the long axis. The side lobes 
are not well suppressed, one being as high as —12 
db. Frequency, 20 kc; test distance, 12 feet; 
depth, 9 feet. (UCDWR CY4 transducer.) 

0 


Figure 47. The directivity pattern of a unit 
designed for sharpness in one direction and 
breadth in the perpendicular shows very narrow 
beam with reasonably good suppression of side 
lobes. Frequency, 90 kc; test distance, 20 feet; 
depth, 9 feet. (UCDWR FE2Z transducer.) 


Figure 48. The directivity pattern of the 
same unit as in Figure 47 but in a plane perpen¬ 
dicular to the preceding. The pattern, as was de¬ 
sired, is found to be much broader and no effort 
was made to suppress the side lobes. Frequency, 
90 kc; test distance, 20 feet; depth, 9 feet. 


Figure 46. The directivity pattern of same unit 
as in Figure 45 in a plane perpendicular to 
the long axis. The short dimension of the crystal 
yields a roughly uniform pattern. Frequency, 20 
kc; test distance, 12 feet; depth, 9 feet. (UCDWR 
type CY4 transducer.) 









































































CALIBRATION DATA 


25 


axis. The pattern will, in general, be down in all 
other directions, because the axis is usually- 
chosen in the direction of maximum intensity 
(and sensitivity). Patterns of a few UCDWR 
units, in specified planes and at specified fre¬ 
quencies, are shown in Figures 41 to 48. The 
factors which influence directivity patterns, and 
their theoretical calculation from given data, 
are discussed in Chapter 4. 

The total energy radiated by a transducer 
actuated by a given signal is a necessary quan¬ 
tity if one wishes to know the efficiency of the 
unit. The total power radiated is the integral 
of the intensity over any sphere completely 
surrounding the transducer, and this is usually 
expressed in terms of the intensity on the axis 
at some standard distance and the integral of 
over the unit sphere. This latter in¬ 
tegral divided by the total solid angle, is known 
as the directivity factor : 


directivity factor = 




in which dco is an element of solid angle. Ten 
times the logarithm of the directivity factor is 
called the directivity mclex. 

Thus the directivity factor of a spherically 
symmetric unit is 1 and its index is 0 db. If the 
pattern of a unit consists of one principal lobe 
which is conical (i.e., axially symmetric) to¬ 
gether with a number of minor lobes, then the 
directivity factor is readily obtained from the 


angle $ at which the pattern is down 3 db: 


directivity factor = sin^ 



This equation assumes the source is uni¬ 
formly driven, in an infinite baffle, etc., but 
is quite adequate for most transducers (see 
Section 4.4). 

For example: a pattern ±10° wide at the 
—3 db points has a directivity factor of ap¬ 
proximately 0.0076 and an index of —21 db. 
If (9 ^ 30° the error in the directivity factor 
caused by setting sin i9 = ^ is less than 10 per 
cent. 


Responses 

There are a number of properties of a trans¬ 
ducer which go under the general name of 
“responses.’’ For a transmitter, the response is 
the intensity, usually expressed in decibels, on 
the axis for any of a number of different con¬ 
ditions of drive, such as constant voltage across 
the transducer terminals, constant current into 
a specified length of specified cable, and out of 
a real amplifier. It is important to notice that 
the response is meaningless unless the condi¬ 
tions of drive are carefully specified. Responses 
are usually given as a function of continuously 
increasing frequency for given conditions, such 
as a current of 10 ma into the unit, or 1 w of 
available power; however, the variation of fre- 



15 20 25 30 

FREQUENCY IN KC 


Figure 49. Transmitter response of the same 
transducer whose patterns were shown in 
Figures 45 and 46. In this case, constant 
voltage is applied to the transducer terminals. 
Test distance, 10 feet; depth, 9 feet. (UCDWR 
CY4 transducer.) 
















26 


GENERAL SURVEY 


qiiency is not essential, as one might, for ex¬ 
ample, give the response at a single frequency 
as a function of the power input, to determine 
whether the system is linear. 

The receiver response of a unit is some meas¬ 
ure of the electric signal for a sound wave of 
given intensity, incident along the axis and for 
specified conditions of termination of the re¬ 
ceiver (for example, open-circuit voltage across 
the terminals, open-circuit voltage at the end 


of some definite length of cable, output at the 
terminals of a built-in preamplifier). 

The transmitter and receiver responses of 
several UCDWR units, under stated conditions, 
are shown in Figures 49 to 51. 

Factors influencing the responses of a unit 
are extremely complicated, depending upon the 
most intimate details of the internal structure. 
The control and improvement of responses 
form an important part of our subject matter. 



FREQUENCY IN KC 

Figure 50. The frequency response as a receiver of a broad band transducer. Here the open circuit 
voltage across the transducer terminals is plotted against the frequency, and it will be noted that between 
57 and 100 kc the I’esponse is uniform within ±214 db. Test distance, 10 feet; depth, 9 feet. (UCDWR 
XCCZ4 transducer.) 


(Vi 

Z 


u 

z 


o 

> 


o 

-I 

bi 

CO 

a 

o 



70 80 90 100 110 120 

FREQUENCY IN KC 


Figure 51. Frequently a transducer is designed for a service in which only a very short length of 
cable, say 2 or 3 feet, is needed, but in order to calibrate it, a much longer piece must be attached. A 
knowledge of the impedance of the cable and transducer permits a correction to be made for the cable. 
This is the open circuit voltage of a transducer, acting as a receiver, measured at the end of 35 feet of 
cable. Near 90 kc, the region of interest, a correction has been applied. The corrected response is about 
10 db higher. Test distance, 20 feet; depth, 9 feet. (UCDWR FE2Z transducer.) 















































APPLICATION 


27 


Impedances 

Still regarding the transducer as a black box, 
we can make electrical measurements at the 
terminals and thus determine the series-equiv¬ 
alent impedance as a function of frequency. 
This impedance, which depends upon the 
elastic as well as the electric properties of the 
system, is merely the complex impedance, at 
each frequency, of the simple circuit to which, 
for the purpose of calculating total current, it 
is equivalent. 

The cable usually consists of three conduc¬ 
tors, the two leads to the crystals, and a shield, 
and the black box is therefore an inert three- 
terminal network. For routine tests, it is cus¬ 
tomary to make measurement on the two leads 
to the crystals, and in this case it is important 
to specify sharply what disposal is made of the 
shield. The most common arrangement is to 
ground the upper end of the shield, leave the 
lower end free, and drive through the leads from 
a transformer whose center tap is grounded. 
The series impedances of several UCDWR units 
are shown in Figures 52 to 55. 

Impedance data, provided the conditions 
under which they are taken are carefully noted, 
serve two valuable functions. First of all, they 
are of immediate and practical importance in 
designing associated electronic systems. Sec¬ 
ond, they supply valuable information for de¬ 
termining the effect of various constructional 
features and for verifying theoretical treat¬ 
ments. 


> ‘ POWER LIMITATIONS 

If the power input to the electric terminals 
of a transducer is increased, the sonic output 
will increase proportionately, at least for a 
while. The practical question now arises: 
“Under various circumstances, what factors 
establish the ultimate upper limit to the power 
which can be radiated?” 

No categoric answer can be given to this 
question, and the best information on the sub¬ 
ject is summarized in Chapter 4. For the pres¬ 
ent, it must suffice to mention a few of the pos¬ 
sibilities. In certain units having a self-con¬ 


tained power supply, the limiting factor will be 
the available power and the efficiency of the 
unit. If unlimited power is available, the limit 
may be set by electrical breakdown within the 
system, by cavitation of the elastic medium, and 
perhaps otherwise. No certain instance is 
known in which the failure of a unit, working 
into water, was caused by dynamic fracture of 
the crystals. Such cases have, it is true, been 
reported; however, it seems likely that these 
failures had other causes such as undetected 
fracture from rough handling followed by elec¬ 
trical breakdown of the damaged crystal due to 
high-voltage puncture. 

The electrical breakdown properties of a unit 
can be improved by careful attention to details 
of construction, and some units have been op¬ 
erated near resonance up to 2,500 v across ADP 
crystals in. thick, or about 4.0 kv per cm. 
These same units are subjected to a routine 
manufacturing test of 4,400 v at 60 c, or about 
7.0 kv per cm. 

Cavitation is, roughly speaking, analogous to 
boiling induced by lowering the pressure. How¬ 
ever, there is some evidence to suggest that 
metastable states are involved, and that the 
previous history of the medium, vaporization 
nuclei, and perhaps other factors make the sub¬ 
ject much more complicated than a simple 
thermodynamic treatment would imply. These 
factors are discussed more fully in Chapters 3, 
4, and 6. 


APPLICATION 

The many and varied applications of crystal 
transducers are not the direct concern of this 
volume, and no attempt is made to discuss this 
subject exhaustively. However, we must men¬ 
tion briefly certain aspects of their application 
in order to understand the nature of the speci- 
flcations toward which we must work. 

Let us first roughly divide all applications 
into listening and echoing. Listening, except in 
very special cases, will be done over a broad 
band, and a suitable receiver must therefore 
have a response which is reasonably flat over 
a considerable range of frequency. The signal- 
to-noise ratio is usually determined by factors 




28 


GENERAL SURVEY 



Figure 52. The complex series equivalent 
impedance, R + j X, oi a. transducer similar 
to that shown in Figure 39; for convenience, 
the negative of the reactance is plotted. The 
two motors are connected in parallel, leading to 
double maxima in the resistance and the nega¬ 
tive reactance. The cable was compensated 
during the measurement by an equal cable in 
another leg of the bridge. (UCDWR CCU8Z 
transducer.) 



transducer whose response was shown in 
Figure 49. Comparison shows that the maxi¬ 
mum resistance and the maximum response 
both occur at about the same frequency. 
(UCDWR XCCZ4 transducer.) 


beyond the control of the designer, and there¬ 
fore it will do no good to increase the energy¬ 
collecting area of the unit beyond a certain 
point; therefore, the size of a receiver is almost 
always determined by the desired directivity 
pattern. Two distinct types of directivity pat¬ 
terns will serve most purposes: one will either 
wish the receiver to collect signals coming from 
any and all directions (thereby also accepting 
noise from all directions), whereupon the direc¬ 
tivity pattern will approach a sphere; or else, 
one will desire to receive signals only from 
within a rather restricted cone, either to im¬ 
prove the signal-to-noise ratio or to determine 
direction, or both. In this last case the direc¬ 
tivity pattern should consist of one main lobe 
with the side lobes reduced as much as possible 
(see Chapter 4). 

Echoing falls into two classes, single and 
band frequency. The first corresponds to trans¬ 
mitting long wave trains (pings), and the sec¬ 
ond includes both frequency modulation and 
short pings. The first requires that the response 
be as high as possible at a single frequency and 


is not greatly concerned with the behavior at 
other frequencies, while the latter requires a 
fairly flat response over a band. Directivity 
patterns will vary from narrow single lobes to 
broad patterns, depending on the particular ap¬ 
plication, how much reverberation and noise 
can be tolerated, etc. 

If a separate transmitter and receiver are 
used, the consideration determining the char¬ 
acteristics of the receiver are similar to those 
in listening, except for the frequency range. In 
some applications, one unit serves both as trans¬ 
mitter and receiver, and here the desired prop¬ 
erties of the transmitter must usually take 
precedence. 

One of the most important properties of a 
transmitter is the amount of power that it can 
radiate either continuously or in intermittent 
pings, depending on the application. The factors 
determining the ultimate power output of a 
unit are quite complicated and, in so far as they 
are understood, are discussed in Chapters 3, 4, 
and 6. 

In some cases, factors outside the control of 













































































APPLICATION 


29 




Figure 54. The complex impedance of the 
transducer whose response was shown in 
Figure 50. Here again it is noted that the 
maximum response and the maximum resist¬ 
ance occur at approximately the same fre¬ 
quency. The second peak is the second 
resonance of the same crystals, as distinguished 
from the first resonances of two different sets 
of ci-ystals shown in Figure 52. (UCDWR CY4 
transducer.) 


Figure 55. Frequently a matching network is 
installed inside the case of a transducer be¬ 
tween the cable and the crystal motor to alter its 
response. This is the complex impedance of the 
unit whose response is shown in Figure 51. It 
will be noted that the maximum resistance now 
occurs at a somewhat lower frequency than the 
resonance shown in that figure. Note also that 
the sign of the reactance has not been reversed 
and that the ordinates are linear rather than 
logarithmic. (UCDWR FE2Z transducer.) 


the designer may determine the type of elec¬ 
tronic gear which must be used; for example, 
certain units may require a self-contained bat¬ 
tery-operated power supply, and in this case 
the efficiency of the unit may become extremely 
important. As a general rule, however, the as¬ 
sociated electronic equipment should be de¬ 
signed to the transducer rather than the re¬ 
verse, since electronic systems are more flexible 
than crystal transducers. If for any reason this 
cannot be done, then it is of the utmost im¬ 
portance to know the characteristics of the 
electronic system with which the transducer 
must work. 

Finally, the type of service to which the 
transducer will be exposed must be taken into 
account. For example, a unit may be required 


to go very deep in the water, whereupon both 
the effects of pressure and of operating the unit 
from a long cable must be taken into account. 
Again, the system may be exposed to shock of 
one kind or another and its mechanical rugged¬ 
ness therefore becomes a matter of prime im¬ 
portance. 

$ ^ ^ 

The purpose of this chapter has been to 
survey briefly some of the general problems 
that arise in the design of crystal transducers. 
The rest of this volume is devoted to a detailed 
study of these problems and attempts to indi¬ 
cate the state of knowledge of the subject at 
present. 

















































































Chapter 2 


THE PHENOMENOLOGICAL THEORIES OF LINEAR DISSIPATIVE 
ELASTICS, DIELECTRICS, AND PIEZOELECTRICS 

By Glen D. Camp 


C RYSTAL TRANSDUCERS involve such a multi¬ 
plicity of physical phenomena that, without 
a theory to correlate experimental data, these 
data would appear as a hopeless jumble. There¬ 
fore, before attempting a detailed study of 
crystal transducers, the theories of elastics, 
dielectrics, and piezoelectrics are here devel¬ 
oped from more basic laws of physics. Elasticity 
is required for itself, to elucidate the properties 
of backing plates, to study viscous dissipation 
mechanisms, etc. Furthermore, a linear dissi¬ 
pative piezoelectric system is a superposition 
of an elastic and a dielectric system, both aniso¬ 
tropic and both occupying the same region, with 
coupling between the two. It is instructive to 
develop the theories of the uncoupled systems 
separately and then couple them. 

The first section is devoted to pure elasticity; 
after developing the general linear anisotropic 
theory, it is specialized to yield the theory of 
isotropic solids and viscous and nonviscous 
fluids. The second section is devoted to the 
theory of linear anisotropic dielectrics. 

In the third section, coupling is established 
between these two systems, yielding the theory 
of linear piezoelectricity. Dissipation of several 
types is introduced, since these phenomena are 
important in crystal transducers. The section 
closes with a study of the symmetry properties 
of Rochelle salt [RS] and ammonium, dihy¬ 
drogen phosphate [ADP]. 

The general reciprocity theorem, and the 
rigorous equivalent circuit for a linear dissi¬ 
pative piezoelectric system, are deduced in the 
fourth section. 

A variational principle, rigorously equivalent 
to the basic boundary-value problem of linear 
dissipative piezoelectric systems, is developed 
in the fifth section. This variational principle is 
used to solve the boundary-value problem for 
rectangular plates of 45° Y-cut RS and 45° 
Z-cut ADP. The solution is obtained first in the 
Mason approximation, and later in a higher 


approximation which includes some additional 
effects of practical importance. 

21 ELASTICS 

The linear approximation to the theory of 
elasticity is adequate for all our work, a very 
fortunate circumstance since the theory be¬ 
comes extremely complicated in higher order. 
The proof that the linear approximation is ade¬ 
quate depends partly on results obtained later 
but will be outlined here. 

Consider a crystal radiating into water at a 
single frequency. The displacement amplitude 
is approximately u = U sin 2kx/1.' in which x 
is measured from the nearest node (or virtual 
node, if there is none in the crystal), I' is the 
wavelength in the crystal, and U is the maxi¬ 
mum amplitude at the real or virtual loop at 
X = XV4, The strain is du/dx which has its max¬ 
imum value 2kU/ 1' ai x = 0. It x — L' is> a face 
at which the crystal is radiating into water, 
then the intensity is 



We can therefore express the maximum strain 
in terms of the intensity, 

l2irU\ {\\/2iy 2irL' 

which is of order 10^^ if the crystal is resonant 
{L' = I'/4) and / = 0.3 w per sq cm per sec, 
the so-called steady-state cavitation level in sea 
water (see Section 4.8). Even if an intensity 
of 10^ times “cavitation” could be achieved, and 
if L' were only ( 1 / 2 )^ of its resonant value, so 
that the crystal was operating four octaves 
away from its resonant frequency, then even 
under these conditions, which are absurdly ex¬ 
treme since voltage breakdown would occur 
long before they could be achieved, the maxi- 


30 





ELASTICS 


31 


mum strain would be only 10~“. It is therefore 
valid, in all applications, to neglect squares and 
products of strain components compared to the 
strain components themselves, the error so in¬ 
curred being considerably less than 1 per cent. 


Physical Principles 

There are two basic physical principles which 
make the formulation of a phenomenological 
theory of elasticity feasible. First, the displace¬ 
ment of the particles in any region is a super¬ 
position of a random thermal agitation together 
with a (relatively) slowly varying and orderly 
displacement. The wavelengths associated with 
elastic vibrations, even for very high frequen¬ 
cies, are very large compared with the average 
distance between atoms® and this permits us to 
treat an elastic medium as a continuum, the 
variation in thermal motion being reflected pri¬ 
marily in the temperature dependence of the 
density, elastic moduli, etc. The dissipation cor¬ 
responding to the conversion of the orderly dis¬ 
placement into thermal agitation in crystals, 
steel, etc., is so small compared to the losses in 
a practical transducer that we neglect it en¬ 
tirely (e.g., the mechanical Q of the resonance 
of a free crystal is greater than 2,000, one drop 
of castor oil on the surface reduces this to a 
few hundred, and in a completed transducer in 
water it is of order 5). Thus we may treat all 
substances as continua, and, furthermore, crys¬ 
tals, steel, and most other metals, glass, etc., 
may be regarded as dissipationless. The latter 
is not true of many plastics such as synthetic 
rubber. 

Second, molecular forces are of extremely 
short range compared to the shortest wave¬ 
lengths encountered, the range being of the 
order of the molecular diameter. This means 
that the elastic effect of these forces, which is 
their average over a time long enough to smooth 
out the random thermal agitation but still short 
compared to the period, is equivalent to a sys¬ 
tem of internal stresses acting across every sur- 

Even at 100 me, the wavelength in air at STP is 
3 X 10^^ cm, while the mean distance between mole¬ 
cules is of order 3 X 10-" . In water the wavelength 
is greater and the separation is smaller, the correspond¬ 
ing figures being 1.5 X 10-3 ^nd 3 X 10-s cm. 


face in the medium. These stresses are gener¬ 
ated by a point (derivative) operator acting 
upon the displacements, whereas if the range of 
molecular forces was comparable with the 
wavelength, a finite-distance (integral) opera¬ 
tor would be involved, enormously complicating 
theoretical treatments. 

The elastic behavior is determined by an 
array of experimentally determined moduli, no 
attempt being made to calculate these elastic 
constants from molecular interactions. In the 
present linear approximation, the maximum 
number of distinct moduli is 21 (see Section 
2.1.4) and any symmetry reduces this number. 
The problem which we must solve is to find the 
motion from these given moduli and from the 
geometry and the density of an elastic system, 
together with interactions with other systems. 
We must first develop the kinematics of con¬ 
tinua. 


“ ^ ^ Displacement and Strain 

We choose a Cartesian'^ coordinate frame xyz 
fixed with respect to the undisplaced position of 
our elastic body.^ To take advantage of tensor 
notation, we shall use with m = 1, 2, 3, inter¬ 
changeably with xijz according to 

Xi = a:, X2 = y, a:3 = 2. (1) 

A displacement takes the material in the im¬ 
mediate neighborhood of to a new point 
given by 

Xm = Xm (Xi, Xo, X3) = Xm + Urn. (2) 

The coordinates of displaced points still refer 
to the original frame, which is fixed with re¬ 
spect to the undisplaced position of the body. 

The three quantities are the Cartesian 
components of the displacement of each par¬ 
ticle, measured from a local origin which is 
just the undisplaced position of the same par¬ 
ticle. This displacement will, in general, be dif¬ 
ferent at each point both in magnitude and di¬ 
rection, and will therefore be a function of xyz ; 
in kinetic problems, it will also be a function of 
time. 

Since the field equations are obtained in tensor form, 
their transfonnation to other frames is a simple matter. 







32 


BASIC THEORIES 


Since to each point in the undisplaced body 
there corresponds a unique point in the dis¬ 
placed body, and conversely, the Jacobian of the 
transformation which generates the displace¬ 
ment, equation (2), must be neither zero nor 
infinite. Neglecting squares of the dimension¬ 
less quantities du^Jdx,^, the Jacobian is 

^ jin 

\x,y,z I \x I 

= !(^”” + ^)i +d!vu, (3) 

in which 8„,„ is the Kroneker delta, 1 if 7n = 7i, 0 
otherwise. 

If we follow the motion of all the material 
that was within some region R bounded by a 
closed surface S, then, owing to the displace¬ 
ment, this material will at a later time be in 
some displaced region R bounded by S. The total 
momentum belonging to this material is then 
given by 


J p(x, y, 2)Xm(x, y, z)dxdydz, (4) 

R 

in which is the velocity of the material at 
xyz, the displaced point which came from xyz. 
Now since there is a unique transformation con¬ 
necting xyz and xyz, this integral can be trans¬ 
formed back into one over the undisplaced 
region R. In this transformation, becomes 
just (xyz), and while there is a correction 
necessary to p, this correction is exactly com¬ 
pensated by the Jacobian factor (see next para¬ 
graph) and, hence, the momentum becomes 

^ pu,ndxdydz. (4') 

R 

By similar arguments, we can calculate any 
other quantity associated with an arbitrary 
region R which is generated by the displace¬ 
ment of a region R (e.g., kinetic energy and 
potential energy). As the only examples of 
cases in which the deviation of the Jacobian 
from 1 must be taken into account, we may 
calculate the change in density and the related 
quantity, the dilatation. By our definition of R, 


it contains exactly the same particles as R, and 
hence contains the same mass 


= m = J" pdxdydz = ^ pdxdydz 



dxdydz. 


(5) 


Now since equation (5) is valid for any arbi¬ 
trary region, we have 


The dilatation, defined as the increase in vol¬ 
ume per unit volume of a very small undis¬ 
placed region [i.e., the limit of (V — F)/E], 
can be calculated directly from equation (6) by 
taking advantage of the equality of masses, or 
as follows: 


V-V = J'dxdydz — J* dxdydz 


= ^ ~ dxdydz. 

R 


(7) 


For a small enough region, the bracket may 
be taken outside of the last integral in equation 
(7), whereupon the remaining integral becomes 
just V, and we have 

lim ~ - 1 ^ div u. (8) 

Thus div u is the fractional increase in the vol¬ 
ume at each point, consistent with equation (6) 
which shows that, neglecting second-order 
terms, it is also the fractional decrease in the 
density. 

We may summarize the foregoing by noticing 
that to calculate any quantity associated with 
a region R or the corresponding displaced 
region R, we may in our approximation ignore 
the distinction between the two regions if the 
quantity is first or higher order (momentum, 
kinetic, or potential energy), and only need to 
observe the distinction if we wish to calculate 
the first-order correction to a Oth-order quan¬ 
tity (mass, volume). 









ELASTICS 


33 


To complete our study of the kinematic 
aspects of displacements, we must now develop 
the well-known result that the most general in¬ 
finitesimal displacement may be regarded as the 
superposition at each point of a rigid transla¬ 
tion, a rigid rotation, and a deformation which 
distorts every small sphere into an ellipsoid. 
The local translation is just u^, and the local 
rotation and deformation are determined by the 
antisymmetric and symmetric parts of the ten¬ 
sor du^Jdx^, respectively. These local displace¬ 
ments fit together to form a displacement of the 

x+dx= x+u+dx+du 



Figure 1. Illustration of displacement and 
strain. 

whole body without fracture precisely because 
they are generated by a continuous vector field 
through its derivative tensor, and, hence, 
there are differential identities connecting the 
local displacements at neighboring points. 

Referring to Figure 1, consider all those par¬ 
ticles which, in the unstrained state, lie on dx 
between x and x dx (subscripts are dropped 
in this discussion, but all quantities are 
tensors). The displacement takes x to x — 
X u and x dx to = x -\- dx u -\- 

du. Thus, the particles on dx are translated by 
u and the translated vector d is rotated and 
extended by du to form dx, according to*" 

c The notation ,n is the commonly used abbreviation 
for cl/dxn. Also, the Einstein convention is used, 
whereby repeated indices are automatically summed 
over, unless the contrary is specifically stated, without 
writing a summation sign. 


dXm = dXn = 4 - Um.n) dXn- (9) 

We now separate „ into its symmetric and 
antisymmetric parts, both of which are tensors: 


u„,,n = Smn + Cl mn 

Umn Unm 

s.. =-2- 



( 10 ) 


The matrix {1 -\- u) in equation (9) differs 
only by the small matrix n from the identity, 
so that neglecting second-order small quantities. 


{1 u) = (1 -j- s -f- ct) 

— (1 + s) (1 + Cl) 

— (1 + o) (1 + S), (11) 

dXm = (5™p + S,np) + Opn) dXn 

= (5mp + Omp) i^pn Sp„) dx„. (12) 

Equation (12) asserts that dx is generated 
from dx by applying the matrices (1 + a) and 
(1 + s), and that the order is immaterial so 
long as we neglect second-order terms. Now 
(1 + a) is the matrix of an infinitesimal rigid 
rotation, while (1 + s) carries a sphere into an 
ellipsoid. The rotation is given by 


9 = h curl u. (13) 

The equation of the ellipsoid, referred to a 
frame parallel to our fundamental frame but 
with origin at x, is obtained as follows: we con¬ 
sider all elements dx of some constant length r 
and ask upon what surface their ends lie after 
the displacement. We already know that their 
initial ends all move by u to x, and their length 
and direction is given by equation (12). Solving 
equation (12) and forming the square of the 
length of dx, we have 

dXm ^ (5„,„ — Um,n)dx„, (14) 

dx„,dxm = r~ 

= dXj, (8,„p - Um,p) {.8mn “ U m,n) dXp 

= dXp{bpn — 2Spn) dXn. (15) 

Equation (15) asserts that a quadratic form 
in the dx is a constant r-, and this is an ellipsoid 
because the principal values of 1 — 2s are all 
positive since all components of s are small 
compared to 1; physically, a hyperbola would 
correspond to rupture. 

Now from equation (15), or more directly 








34 


BASIC THEORIES 


from equation (12), after discarding the rigid 
rotation (since it does not change the length of 
dx), we see that the principal values of r/r, the 
ratio of the length of dx to that of dx, are 1 plus 
the principal values of s, and the extensions in 
the three principal directions are therefore 

^ _ r = r(principal values of s). (16) 

Thus, the principal values of s are the principal 
strains and is therefore called the strain ten¬ 
sor. Anticipating a later application, it should 
be emphasized that only s contributes to the de¬ 
formation of an element dx, and, hence, only 
this part of iq,,,, can occur in the potential energy 
density. 

In nearly all works on elasticity, the strain is 
specified by a matrix whose diagonal ele¬ 
ments are identical with but whose off-diag¬ 
onal elements are just twice as large. Unlike s„,„, 
is not a tensor,--and this causes dissym¬ 
metry in the equations of motion (see Section 
2.1.3) and complicates the transformation of 
these equations to other frames of reference. 
Despite these disadvantages, long usage has 
firmly established the in elastic theory, and 
it is necessary to know the distinction between 
them and the strain tensor s„„,. 


Stresses; Equations of Motion 

As remarked in Section 2.1.1, the forces with 
which one part of an elastic body act upon an¬ 
other have a very short range and therefore, 
on a macroscopic scale, are equivalent to a sys¬ 
tem of stresses over every surface inside the 
body. The force on an element of area dS must 
be proportional to dS and, except in fluids, must 
also depend upon its orientation. Every surface 
element dS has two normals pointing in oppo¬ 
site directions and we shall say that dS belongs 
to that part of the material lying on the side 
toward which the normal is taken. Now choose 
an element whose normal points in the direction 
of the positive x axis and call the components 
of whatever force acts upon it, 

-P.4S, -Py4S, -P,4S. (17) 

The minus signs are chosen to agree with con¬ 
ventions used in most works of elasticity and it 


should be noted that the components of the 
stress normal to the surface, is positive for 
a tension and negative for a pressure. The tan¬ 
gential components of the stress and P^^ 
will, in general, vanish only for a fluid. When 
we say that the above are the components of the 
force acting upon dS, we mean that they give 
the force with which the material on one side 
acts upon the material on the side to which dS 
belongs. Equation (17) so far applies only to a 
dS belonging to the material on its positive side, 
but the force on the material on the negative 
side is just the reaction and can therefore be 
obtained by reversing the sign of each compo¬ 
nent. 

Now take two other elementary surfaces of 
the same area dS, one with its normal pointing 
in the positive y direction the other in the posi¬ 
tive ^ direction. Call the forces on these ele¬ 
ments 

-P.ydS, -PyydS, -P^ydS, 

-P,4S, -PydS, -P^JS. (18) 

The unit normals to these three surfaces are 
respectively (1,0,0), (0,1,0), and (0,0,1). Let¬ 
ting stand for any of these normals, all the 
above relations can be condensed into the single 
expression 

— PrsnsdS. (19) 

Furthermore, by letting take values 

( — 1,0,0), etc., equation (19) also gives the re¬ 
actions correctly. 

This looks like the contraction of a second- 
rank tensor with the unit vector which deter¬ 
mines the orientation of an arbitrary dS and, 
since there is nothing special about the direction 
of the coordinate frame, we suspect that this is 
a general relation. This is, in fact, true, and it 
can be shown that unless the stress on an arbi¬ 
trary element is given by the above formula for 
all orientations of dS, the acceleration of an ele¬ 
ment of volume would be infinite, since only 
then will the total force over the surface of a 
small region go to zero like the volume instead 
of the area.--'^ 

That P,^ is actually a tensor follows from the 
fact that —P,,,%, is the force per unit area, a vec¬ 
tor, for arbitrary n/, that is, contracting P,.^. 
with an arbitrary vector gives a vector, and this 




ELASTICS 


35 


is only true of a tensor. Furthermore, it can be 
shown that unless is symmetrical, the couple 
acting on a small closed region will go to zero 
only like the cube of its linear dimensions while 
the moments of inertia go to zero like the fifth 
powers of these dimensions and, hence, the 
angular acceleration will be infinite.-"* Thus, we 
conclude that the stresses are represented by a 
symmetric second-rank tensor. We have not yet 
shown how to calculate its components, but we 
know that it can depend only on the strain ten¬ 
sor, the symmetric part of together with 
material constants. 

The resultant of all these surface forces, act¬ 
ing over the closed surface S bounding a region 
R, will be equivalent to a total force, acting on 
the material in R, of amount 

Prsn,dS dV (^3 inward), (20) 

s i? 

in which the transformation from a surface to 
a volume integral is made with Gauss’s flux- 
divergence theorem. Since equation (20) is 
valid for any arbitrary region R, we see that 
the resultant of the surface stresses is a body 
force of (dP^ydxJ per unit volume. 

The equation of motion of an arbitrary in¬ 
finitesimal element of volume may now be ob¬ 
tained by taking the time derivative of the total 
momentum, equation (4), again taking advan¬ 
tage of the arbitrariness of R to drop the inte¬ 
gral sign, 

(pdV)ilr = ( ^ yv + B4V, (21) 

in which Br is the force per unit volume arising 
from any external fields (e.g., gravitation). 
These three equations become propagation 
equations for the displacement components 
as soon as we have established a relation be¬ 
tween the stress and strain tensors. 

Energy Density; Generalized 
Hooke’s Law 

Let us now consider a body which is in 
equilibrium under the action of surface forces 
and, perhaps, a body force density B^, both 
arising from some external agency. If we now 


cause the surface and body forces to vary in 
such a way that the displacement receives a 
small arbitrary variation, we can, on the one 
hand, calculate the work done by the external 
agency and, on the other, the increase in po¬ 
tential energy, the latter provided we assume a 
potential energy density. 

The deformed body will possess a certain 
amount of potential energy which is distributed 
throughout it with some volume density W. This 
potential energy density function W must de¬ 
pend only upon the strain tensor together 
with certain phenomenological constants char¬ 
acteristic of the material, and it must be a posi¬ 
tive definite function of this strain tensor (i.e., 
vanish only if every component of the strain 
tensor vanishes, and be positive otherwise). 
The simplest function of this type is a quadratic 
form in the and since these are small quanti¬ 
ties, all higher-order terms may be neglected. 
We therefore have 

W = (22) 

In equation (22), the stiffness moduli must 

form a fourth-rank tensor, because W must be 
a scalar for arbitrary values of the tensor ; 
this determines the manner in which they 
change when referred to a new coordinate 
frame (see Section 2.3). Since the are sym¬ 
metric in pg, the must be symmetric under 
an interchange of p with q and also r with s, 
so that only six values of the pairs 'pq and rs can 
give distinct components. Furthermore, ex¬ 
change of the pairs pq and rs does not alter W, 
and hence must not alter the tensor. It therefore 
has at most as many distinct elements as a sym¬ 
metric sixth-rank matrix, namely 21. Any sym¬ 
metry will reduce this number. 

Now let us suppose the body, or any region 
within it, to be in static equilibrium under the 
combined action of surface forces of amount F,. 
per unit area and body forces of amount Br per 
unit volume. The work done by these forces, 
corresponding to an arbitrary infinitesimal 
variation of the displacement, must be equal to 
the increase in potential energy, and we there¬ 
fore have 

^Frauds + ^BrbUrdV = 8sp^V. (23) 

s R ^ 





36 


BASIC THEORIES 


For equilibrium, the surface forces must bal¬ 
ance those arising from the stresses, F,. = Py^n^ 
(outward normal). This enables the surface in¬ 
tegral to be replaced by one over the volume, 
giving 

/(+ i 

R 

The external body forces By combined with the 
internal dPyJdx^ must be in equilibrium, other¬ 
wise, by equation (21), there will be an accel¬ 
eration, and hence the first integral in equation 
(24) is zero. Because of the symmetry of P,.„ 
6 m,. ^ may be replaced by 6s,. ^ and hence, since the 
region R is arbitrary, we are left with 

D dW 

^ = Cp,rsSr.. (25) 

Equation (25) states that the stresses to first 
approximation are linear functions of the 
strains, which is a generalization of Hooke’s 
law. We could have started from this assump¬ 
tion and deduced the potential energy density. 
The symmetry would then have come from the 
integrability conditions. 

If we insert the above values of the stresses 
into the equations of motion, equation (21), we 
have three field equations which govern the 
propagation of the displacement at all interior 
points. If, as is usually the case, the By are 
zero, then equation (21) represents the propa¬ 
gation of waves in a crystalline medium having 
no sources. 

To get a completely determinate system, we 
need know only two more things: the numerical 
values of the elements of the stiffness tensor 
and the conditions which exist at the bounda¬ 
ries, the latter discussed in the next section. The 
determination of the elastic moduli for any 
particular substance, together with their trans¬ 
formation to other axes, is discussed in Section 
2.3 on piezoelectrics. Here we shall have other 
phenomenological tensors that require consid¬ 
eration, and shall also have to discuss their 
transformation to the matrix form, more con¬ 
venient for detailed calculations as distin¬ 
guished from general theoretical treatments. 


^ Boundary Conditions 

To obtain a determinate problem, the equa¬ 
tions of propagation must be supplemented by 
boundary conditions which may take a variety 
of forms. These are of the utmost importance, 
since to falsify the boundary conditions corre¬ 
sponds to assuming the existence of external 
forces which are not actually operating. The 
principal types of boundary conditions are as 
follows. 

1. Block. Over some external surfaces of the 
system it may be assumed that some rigid fas¬ 
tening prevents all displacement. This is, of 
course, an idealization since nothing is rigid if 
we go to a high enough frequency, but it may be 
a satisfactory approximation in certain cases. 

2. Free. If a surface is in contact with air the 
boundary condition is that the stresses over 
that surface shall be zero. 

3. Driven. A surface may be exposed to cer¬ 
tain external driving forces (receiver). The 
boundary condition in this case is that the ex¬ 
ternal forces shall match those arising from the 
internal stresses. 

4. Impedance. In later work w^e will have 
occasion to assume impedance boundary condi¬ 
tions and, even though the numerical values of 
the (usually complex) surface impedances may 
be very difficult to evaluate, we shall find this 
to be a very valuable means of representing cer¬ 
tain physical situations. If a part of the exter¬ 
nal surface looks into an inert medium to which 
it is elastically coupled then we may expect re¬ 
actions which are linear functions of the dis¬ 
placement and its time derivatives, correspond¬ 
ing to dissipative and reactive loads. We need 
only find the steady-state solutions in the fre¬ 
quency range of interest, since other solutions 
can be built up from these by Fourier integrals. 
We therefore introduce a normal and a tangen¬ 
tial specific acoustic impedance,"^ and match the 
normal component of the internal stress with 
the normal impedance times the normal ve¬ 
locity, and the tangential component with the 
tangential impedance times the tangential ve¬ 
locity. 

The tensor formulation of boundary conditions in¬ 
volving a tangential as well as a normal impedance is 
discussed in reference 4. 





ELASTICS 


37 


It should be noticed that the impedance 
boundary condition includes the blocked and 
free conditions as special cases, corresponding 
to infinite and zero impedance respectively. 


Isotropic Solids 

The isotropic solid is a special case defined 
by the condition that there are no preferred di¬ 
rections within it. Its stiffness tensor must 
therefore be the same in all Cartesian frames. 

The transformation of the various material 
tensor under rigid rotation of the Cartesian 
axes is discussed in Section 2.3.3, where the 
piezoelectric and dielectric tensors are treated 
in addition to the elastic tensor. Using the 
methods of that section, together with the re¬ 
quirement of complete symmetry, one readily 
concludes that the stiffness tensor for an iso¬ 
tropic solid is 

Cpqrs = y^dpqdrs + + 8qr8ps), (26) 

in which A and p have been chosen so as to agree 
with the notation of Love.-*^ 

In this section we are concerned with pure 
elasticity (as distinguished from piezoelectric¬ 
ity), primarily in rods and plates. The theory 
of rods is contained in that of crystal plates (see 
Section 2.5.3), and is most conveniently handled 
by the equivalent circuit representation as 
given in Chapter 3. One merely short circuits 
the condenser and uses the simplifications aris¬ 
ing from the isotropic given above; this 
amounts to setting the thickness and width 
Poisson ratios equal, and the shear Poisson 
ratio Eg to zero. 

The pure thickness vibration of plates is iden¬ 
tical in theory to that of rods, the only differ¬ 
ence being that the thickness modulus replaces 
Young’s modulus, the former being about a 
third larger for steel. For calculating the reso¬ 
nant frequency, this is important; however, for 
calculating a steel backing plate to be a good 
block, it is of no consequence, since the charac¬ 
teristic impedance of steel is so high that even 
a one-eighth wave plate is a good block. 

However, plates also vibrate in flexure, and 
here the situation is complicated and of prac¬ 
tical importance since any nonuniformity in a 


crystal motor will tend to excite these flexural 
modes. This matter is discussed in Chapter 3. 


‘ Nonviscoiis Fluids 

Field Equations ; Boundary Conditions 

In a nonviscous fluid, the stress arises solely 
from the dilatation, the rigidity modulus u being 


zero, 

P pq — ^Ur,r8pq = P^pqf (27) 

p — — 'KUr,r = — X div u. (28) 

The propagation equation is 

pUr = Prs.s = — P.r = X (diV U),r (29) 

pii = — vp = Xv div u. (30) 

The boundary conditions at any surface are 

Ppqriq = - pUp = Fp, (31) 


in which is the force per unit area exerted on 
the boundary surface by external agencies. 
They are of course contradictory unless is 
normal to the surface, since a nonviscous fluid 
cannot support a tangential stress. 

Steady State 

Only the steady-state problem is of interest in 
this volume since, even if one could handle the 
general time-dependent problem, its solution 
wmuld be useless since only the steady-state 
motion of crystals is considered. The usual 
method of Fourier integral representation is, of 
course, available for the study of short pings. 

There are two steady-state conventions in 
common use. One uses exp(za)t) for the time 
factor, as in most works on electric circuit 
theory. A plane wave traveling in the positive 
X direction, exp(zo)t -(- ikx), therefore has a 
propagation vector which points in the opposite 
direction to that in which the wave travels. To 
avoid this, Morse-^ and others use exp (—wot) 
whereupon k is positive for a wave traveling in 
the positive direction; likewise, Hankel and 
Heine functions of the first kind represent out¬ 
going waves. This latter convention is followed 
in this volume when discussing elastic waves in 
fluids, whereas the positive time-exponent con¬ 
vention, exp(4-ta)t), is used when discussing 
crystals, etc., where the equivalent circuit rep- 




38 


BASIC THEORIES 


resentation urges conformity with electric cir¬ 
cuit theory. If the positive exponent is used 
(crystals), an inductive or mass reactance is 
positive, a condenser or spring reactance is 
negative; the use of the negative exponent 
(fluids) reversing the signs of these reactances. 
Therefore, when combining results using two 
different conventions, it is necessary to make a 
qualitative check to be sure that the reactances 
have the proper signs. 

The propagation equations for the pressure 
are readily obtained by taking the divergence 
of equation (30), and using equation (28), 

(V“ + ^-)p = 0 (except at a source), (32) 

= $ ^ = (7)- 

The pressure then serves as a velocity potential. 


The boundary conditions which must be ap¬ 
pended to yield a determinate problem may take 
a variety of forms. The simplest is that the 
pressure shall be zero over a given surface 
(free surface) ; another is that the normal com¬ 
ponent of velocity shall be zero; these are both 
included in the impedance boundary condition 

p = Z(v • n), (35) 

in which Z is the normal specific impedance. 

In addition to the boundary conditions at in¬ 
ert surfaces, there must also be a source or 
active surface, otherwise all amplitudes are 
zero. 

Neumann Bound ary-Value Problem ; 
Green’s Functions 

In the present state of knowledge of crystal 
transducers, the mechanical coupling between 
crystals and the impedance loads on their faces 
are very imperfectly known. There is therefore 
nothing to be gained at present by trying to 
take account of the impedance presented to the 
individual crystals by the radiation field even 
if one knew how to do this extremely compli¬ 
cated calculation. One must rather try to deter¬ 
mine the general rules governing the depend¬ 
ence of this impedance on crystal spacing and 
velocity distributions over the face. 


The most important practical problem in 
transducer design studies is therefore a Neu¬ 
mann problem*^; to find the pressure in an in¬ 
finite fluid region when the normal component 
of this velocity is given at each point on a closed 
surface (the transducer). At the present time, 
no general solution to this problem is known 
(even for the simplest of all surfaces, a sphere). 
Nevertheless, it will be seen that valuable re¬ 
sults are obtained by using certain approximate 
solutions; these approximations are the basis 
of all present calculations of directivity pat¬ 
terns and the radiation impedance seen by a 
transducer. 

The problem to be solved is 

(y- + ^“)p =0 (outside the transducers), (36) 

/^\ 

_ \dn } (given on transducer), (37) 
ikpc 

p outgoing, like D {6, </>) exp as r -> «. (38) 

r 

There are two general attacks on this prob¬ 
lem : If the transducer surface is simple enough 
(sphere, long cylinder), one can obtain a for¬ 
mal solution in the form of an infinite series of 
characteristic functions. This method is ap¬ 
plied in Chapter 4 to get some very useful ap¬ 
proximate results. The other is the Green’s 
function method which leads to the rigorous 
form of the Huygens-Fresnel principle to be 
discussed here. 

A Green’s function of the above problem is 
defined as any function of the form 


2) = go(l,2) + IF(1,2), 

(39) 

^o(l, 2) = exp ikri 2 

(40) 

rn ’ 


rn = I r 2 - ri 1, 

(41) 


in which IF (1,2) is a solution of equation (36) 
at every point outside the transducer; that is, it 
has no singularities (sources) outside the 
transducer (and hence must have some inside, 
since no function can satisfy equation (36) in 
all space). Physically, a Green’s function repre¬ 
sents the field produced by a point source (first 
term on right) in the presence of some unspeci¬ 
fied kind of reflection from the transducer (and 
perhaps other surfaces in the general problem. 






ELASTICS 


39 


but not here, since the transducer is assumed 
to be in an infinite medium). 

Outside the transducer, ^( 1 , 2 ) satisfies 

(Vi + ^-)^(1,2) = 0, except at ri = (42) 

Multiplying equation (42) by p(l), and equa¬ 
tion (36) (evaluated at Fi) by ^(1,2), and tak¬ 
ing the difference, one has 

divi [pa)v,g{l,2) -g(l,2)ViP(l)] =0, (43) 
this being valid at every point outside the trans¬ 
ducers and outside a small sphere surrounding 
To. Converting this to a surface integral over the 
transducer and the sphere, and letting the 
sphere shrink to zero, one has the rigorous 
Huygens-Fresnel principle 

transducer 

(44) 

This is valid for all Green’s functions, includ¬ 
ing the known o',, ( 1 , 2 ) ; however, it is not a 
solution of the problem because it contains not 
only dp/cln, known on the surface, but also the 
unknown p(l). 

If fir (1,2) is the rigid Green’s function, that 
is, if IF ( 1 , 2 ) represents the waves reflected 
from a rigid transducer, then 

— 3 ^- ’ -- = 0 (on transducer), (45) 

uTli 

and the unknown p(l) falls out [but one now 
has which is unknown], 

p(2) = - ( -ff ) / dS,gr(l,2)v,.{l). (46) 

No entirely satisfactory approximation to 
fir,.( 1 , 2 ) is at present known, even for a sphere, 
despite the fact that this subject has been in¬ 
tensively studied by Rayleigh and others,"’ ® but 
some very useful semiquantitative conclusions 
can be reached by the study of special cases. 

One would like to be able to calculate the 
pressure corresponding to a given velocity dis¬ 
tribution over the transducer surface: at points 
on the surface, to get the impedance seen by 
the transducer at each surface point; and at 
points far aivay compared to the wavelength or 
the longest dimension of the transducer, which¬ 
ever is larger, to calculate the directivity pat¬ 
tern and as an alternate method of estimating 


the order of magnitude of the resistive part of 
the impedance. 

Most transducers must have dimensions of 
the order of a few wavelengths, in order that 
suitable directivity patterns can be obtained. 
Thus, they fall in the critical region between the 
long and short wave limits; nevertheless, it is 
helpful to study these limiting cases. 

First consider (/,.(1,1') for two points ri and 
r/ both on the surface. This surface-surface 
Green’s function yields a formal expression for 
the impedance, 

I 

It will be shown in Chapter 4 that, in the long 
wave limit, only the volume-velocity radiates 
appreciably, and hence only the average normal 
velocity sees an appreciable resistive component 
of radiation impedance. We are not interested 
in the reactive component because it will merely 
shift the resonance frequency slightly, and 
hence we are not interested in the surface-sur¬ 
face Green’s function in the long wave limit. 

In the short wave limit (that is, wavelength 
small compared to the radius of curvature or 
the smallest distance on the surface in which 
the normal velocity changes appreciably, which¬ 
ever is smaller), each element of surface will 
behave essentially like an infinite plane. The 
impedance must therefore approach a pure re¬ 
sistance of QC and hence the Green’s function 
evidently approaches (—iji/ikoc) times a delta 
function of the two surface points. 

The special case of a sphere throws light on 
how short the wavelength must be before this is 
a good approximation. Let the pressure and the 
surface velocity be expanded according to 


Lp- 

rhn(kr) 

\_hn{ka) 

J Pn{cos 6) exp im4>. 

(48) 

v(a) 


,P^'(cos 6) exp imf). 

(49) 


hn{Z) = 


(50) 


Then the pressure and velocity coefficients are 
connected by 


Pnm = pCtn(ka)V„„„ (51) 

f,(.) = If, (52) 

in which, because of equation (51), the are 
called modal impedances in oc units; it should 










40 


BASIC THEORIES 


p(r) = pcj^^„ika)un 


\ Pn exp imcf). (48') 


be noticed that they are independent of the azi¬ 
muthal quantum number. Using equation (51) 
to replace the in equation (48), one has 
~ K{kry 
Jin{ka)_ 

In the short wave limit, ka is large, and it can 
be shown that the as n increases, are very 
close to unity up to n—.ka, where they rise very 
suddenly to a large maximum in absolute value, 
after which they fall to zero very rapidly. Now 
suppose that the velocity distribution is such 
that the are negligible beyond n = ni < ka ; 
then the sum can be broken at 7ii and the all 
set equal to 1, whereupon, on the surface, one 
has 

_p(a) ~ pcunia), Til < ka. (53) 


The condition for validity of this result can 
be expressed in very simple physical terminol¬ 
ogy, related to similar results in physical optics. 
If the normal velocity can be built with negli¬ 
gible error using n"s only up to n-^, this means 
that it changes by an appreciable fraction of it¬ 
self in a distance of the order of 2na/ni or 
larger; the above inequality can be written as 


2xa 

Ui 




(54) 


and we see that on a sphere whose radius is a 
few wavelengths, the details of the velocity ai'e 
resolved if the 'wavelength is small compared to 
the distance along the surface in ivhich the ve¬ 
locity changes appreciably. By replacing 7'adius 
by iridius of curvature one has a result ex¬ 
pressed in general physical terms which make 
no reference to the specific properties of the 
sphere, and one therefore believes that the above 
is general; in confirmation of this, one finds the 
same result on a cylinder (see Section 4.2). 

While the foregoing does not yield the radia¬ 
tion resistance in the intermediate cases, it 
furnishes a very useful criterion for determin¬ 
ing whether the full pc loading will be achieved 
in a particular case and how closely the individ¬ 
ual crystals must be packed to avoid their reso¬ 
lution, with resultant scallop-edged patterns. 
These results are discussed more fully in Chap¬ 
ter 4; for the present, our conclusion concerning 
the surface-surface Green’s function is that, as 
the wavelength decreases, it approaches a delta 
function, being very large at r/ = rj and falling 


rapidly to zero after [r/ — ri| exceeds a wave¬ 
length or so, provided the radius of curvature 
of the surface is large compared to the wave¬ 
length. 

Next, consider fir,.(l,2) for an ri still on the 
surface but ra very far away compared to the 
wavelength or the largest dimension of the 
transducer. Call this surface-distant Green’s 
function 

^.(1,2) = r^o(l,2), (55) 

in which x depends upon wavelength, the geom¬ 
etry of the surface, and upon the direction of ro 
but negligibly upon its magnitude (this latter is 
the essential definition of “distant”). If we 
knew the function x, two useful calculations 
could be made: the directivity pattern and the 
total power radiated; from the latter, one can 
estimate the radiation resistance (see Chap¬ 
ter 4). 

In the long wave limit, one readily concludes 
that a approaches unity (its value for infinite 
space) by taking advantage of the symmetry of 
5 rr(l, 2 ) : instead of regarding ^'^(1,2) as the 
pressure at ro caused by a source on the rigid 
surface at ri, one regards it as the pressure at 
ri caused by a source at r 2 . Then by doing a 
simple scattering problem, one concludes that 
an obstacle small compared to the wavelength 
will very slightly perturb the incident wave 
field even on the surface of the obstacle (one 
should picture long waves slowly compressing a 
minute obstacle). 

In the short wave limit, each point on the sur¬ 
face will act roughly like a rigid plane tangent 
at that point, provided the points are geomet¬ 
rically visible from one another. 

Thus, the surface-distant Green’s function 
approaches the two limits given by equation 
(55) and 

Long wave limit: r ~ 1, (56) 

Short wave limit: r ~ 2 (if ri geometri¬ 
cally visible from 

r2), 

— 0 (if not). (57) 

As previously remarked, transducers tend to 
fall in the intermediate region between the 
long and short wave limits: if Z) is some char¬ 
acteristic dimension of the motor, directivity 
considerations usually make kD, or 2kD/1, any- 







ELASTICS 


41 


where from 1 up to 10 or 15. A careful study^ 
of the Green’s function of the rigid sphere, 
extending Rayleigh’s" work, indicates that in 
this intermediate region the surface-distant 
Green’s function is given better by the Kirch- 
hoff directivity factor than by either equation 
(56) or (57), 

Intermediate region: r ~ (1 + cos a), (58) 

in which a is the angle at ri between the out¬ 
ward normal and (ra — rj. It should be noticed 
that equation (58) is a compromise between 
equations (56) and (57). 

The foregoing results are applied to the cal¬ 
culation of radiation resistance and directivity 
patterns in Chapter 4. 

Energy Density and Flux 

The total elastic energy density and flux in 
a nonviscous fluid is 

£ = 4 - + = 

Sp = — PpgUg = - pc-Up (div u). (60) 
The rate of increase of energy per unit volume 
should be equal to the convergence of the energy 
flux. One has 

— div S = pc- (u • Y div u + div li div u), 

■ f ■ '»« 

in which A div u is replaced by means of the 
propagation equation. 

We must see what form these quantities and 
the conservation law, equation (61), take in 
steady state. First, an ambiguity from a strict 
mathematical viewpoint must be noted: In 
steady-state formalism, long usage has caused 
three distinct quantities to be represented by 
a single symbol, relying on context to distin¬ 
guish them. Taking the pressure p as an exam¬ 
ple, this represents: 

1. The actual time varying pressure, in ex¬ 
cess of hydrostatic. 

2. The complex time varying function p' 
whose real part is p. 

3. The complex amplitude of p', the quan¬ 
tity obtained by dropping the time-factor 
exp(—i(x>t), say p", a function of spatial varia¬ 
bles only. 


These quantities are related by: 

p' = p"exp{ — iut), (62) 

p = 6{p' = (Rp" cos cot -f Op" sin cot, 

(63) 

= I p" I cos (cot — phase p"). 

The advantages arising from having a single 
symbol for these three quantities far outweighs 
the formal mathematical objection that they 
are actually distinct. However, one must be 
careful in forming second-degree quantities 
such as occur in the energy density and flux. 

The energy flux in steady state is, from equa¬ 
tion (60), 


S = ((Rp)((Rv), 

/ VP_ _ VP_\ 
= (p + p) \ ikpc ikpc I 
4 


(64) 


= (—] 
\ Aikpc I 


(pvp - PVP + pvp - pvp). 


The last two terms go like exp(—2{(joO and 
exp (2i(x)t), respectively, and their time aver¬ 
age is therefore zero. One is not ordinarily 
interested in the rapid variations in energy 
density and flux, and hence one keeps only the 
first two terms which are constant in time, 
using the same symbol. 


® ( 2ikpc 1 

1 3PVP, 

(65) 


1 3(vp • YP + PY'p), 

(66) 

/ 1 ^ 


\ 2ikpc j 

1 3(YP • YP - ^^IpI’). 



The first term is pure real; the second is also 
real if k is real, as it is in the absence of ab¬ 
sorption, and hence div 8 = 0. This is con¬ 
sistent with time-averaged conservation, since 
the time average of the energy density is of 
course independent of time. 

The energy density is, from equation (59), 

„ p((Rv)2 ((Rp)2 

= p L \ ikpc jj (p + p)2 
2 ^ 8pc- ’ 

_ vp - VP , IpI^ 

Ak^pc- ' 4pc- ’ 


(67) 

( 68 ) 













42 


BASIC THEOKIES 


In a plane traveling wave, vp = ikp, in 
which case 


E = 


1 ^^ 

2pc- 




plane 

traveling 

wave, 


in which, for convenience, the time-averaging 
symbol has been dropped. 

If a sound field is caused solely by outgoing- 
waves from a single transducer, as is the case 
in discussing the directivity pattern and energy 
radiated, both at distant points, then the radius 
of curvature is so large compared to the wave¬ 
length that equations (67) and (68) are ap¬ 
plicable with negligible error. This is readily 
proven, and the error estimated (it is of order 
l/r), by recalling that the entire field may be 
regarded as a superposition of terms of type 


hn{kr)P'^'{cos 6) exp (im4)). (71) 


Differentiating this with respect to r and using 
the recursion and asymptotic formulas, one finds 
that 


provided n < kr. This latter condition means 
that one is far enough out that the sharpest 
lobe, subtending an angle of order 'Zic/n, has 
an arc distance along the wave front, 2:ir/n, 
large compared with the wavelength. 


^ « Viscous Fluids 

Our interest in viscous fluids arises from two 
possible dissipation mechanisms within a trans¬ 
ducer. The first is the generation of viscous 
shear waves by tangential motion of crystals. 
As will be shown below, these waves are ab¬ 
sorbed in a very short distance (a fraction of 
a millimeter in castor oil) ; hence, they enter 
the theory as a tangential impedance, primarily 
on the lateral faces of crystals (see Section 2.5). 
The second is the partial conversion of longi¬ 
tudinal to shear waves, with complete absorp¬ 
tion of the latter, at reflecting boundaries. While 
these mechanisms cannot be analyzed in detail 
because of the complicated interior geometry of 


transducers, the order of magnitudes of the 
effects to be expected are deduced below. 

Steady-State Boundary-Value Problem 

The stress in a viscous fluid^*’ has tangential 
as well as normal components, the latter de¬ 
pending upon the rate of strain, but otherwise 
having the same structure as that for an iso¬ 
tropic solid, 

P pq — Xdpqllr.r + ^'^p(Ar,T + A A p,q + Uq^p). (73) 
It is customary to assume that V, the dilata- 
tional viscosity, is zero; however, Rayleigh'^ has 
called attention to the fact that the experi¬ 
mental basis for this assumption is scant. 

The steady-state propagation equation is® 

-pco-u = [X-fco(X' + m')] U div u - fcop'(74) 

and the boundary conditions are, as in any 
elastic system, the equality of the flux of the 
stress with the external forces. 

Introducing the usual scalar and vector po¬ 
tentials 


u = u</) + curl A (div A = 0), (75) 

the above propagation equation separates, yield¬ 
ing 


(V- + - 0, (76) 

X. = X - + 2/), (77) 


(V- + k'‘)A = 0, (78) 



The divergence condition on A, applied to a 
plane wave, shows that A represents trans¬ 
verse waves; similarly, 4 > represents longitudinal 
waves. The attenuation of longitudinal waves 
is negligible in the short distances involved in 
a transducer, so that we shall drop the imagi¬ 
nary part of k; it should be noted that this 
causes the dilatational viscosity to fall out, but 
this does not necessarily mean that it is unim¬ 
portant in transmission over large distances. 

e The time factor is taken as exp( —uof). See Section 
2.1.7. 





ELASTICS 


43 


The propagation vector for the transverse 
waves is 


k' = (1 + i). (80,1 

This has a phase of 45°, which means that 
a plane wave is down to (l/e)th value in the 
(l/2jt)th fraction of a shear wavelength. This 
(l/e)th absorption distance is the reciprocal of 
the imaginary part of k', 



which, for castor oil at 20 C and 10 kc, is only 
0.02 cm. 

Tangential Impedance 

From the very short wavelength of shear 
waves we see that any plane a few millimeters 
in its smallest dimension may be regarded as 
infinite when considering specific tangential im¬ 
pedances. Thus the specific tangential imped¬ 
ance imposed on a crystal face separated from 
another surface by a viscous fluid may be esti¬ 
mated by considering a pair of infinite parallel 
planes, one of which is oscillating uniformly. 
Let the plane z = Zi have a tangential displace¬ 
ment amplitude Ug, the plane z = 0 being at rest 
and the space between filled with a viscous fluid. 
Then 


A = (0, To, 0), 

(82) 

. U 0 cos k'z 

^ ” k' sin k'Zy 

(83) 

Pi 3 = — ioofiUok' cot k'Zi, 

(84) 

{Pn)z, 

~ (-fco(7o)’ 


\ Zi j \ tan k'zi I 



For very small Zi, of the order of 10"-^ cm, this 
is a resistance of the order of a tenth of 

the radiation resistance, and since the lateral 
faces have a much greater area than the radi¬ 
ating faces, can be extremely important. A 
clear-cut example of this is found in an experi¬ 
mental UCDWR CY4 type transducer (see 
Chapter 6). If Zi is one or two shear wave¬ 
lengths, is an impedance (p'/L') (1 — i), 
which has equal resistive and mass-reactive 
parts. Thus the resistance is [i/zi until Zi be¬ 


comes of the order of the (l/e)th absorption 
distance and is then constant at (p'/L')- The 
latter corresponds to a plane working into an 
infinite medium. 


Reflection Conversion 

A longitudinal wave obliquely incident upon 
a reflecting surface will produce tangential mo¬ 
tion near this surface, and thus a part of the 
incident energy will be converted to viscous 
shear waves, which will be absorbed in a short 
distance. 

To get some idea of the importance of this 
process, consider a plane longitudinal wave in¬ 
cident upon a rigid plane y = 0, at the angle 6. 
Reflected longitudinal and shear waves will be 
created, and these, combined with the incident 
wave, must make the normal and tangential dis¬ 
placement zero. 

The potentials are 

^i„c = exp ik(x sin 6 — y cos d), (85) 

(pret = r exp ik{x sin + y cos dr), (86) 

Aref = 0,0,A exp i(ax + /3y), 

(a^ + = k'-^). (87) 

The displacements on the plane y = 0 are read¬ 
ily calculated to be 

Ui = 

ik(sm 6 exp ikx sin 6 + r sin dr exp ikx 

sin dr) — ijSA exp iax, (88 ) 

U-2 = 

— ik(cos d exp ikx sin 0 — r cos dr exp ikx 

sin dr) + iaA exp iax. (89) 
These must both be identically zero in x, and 
hence the exponents must all be the same. This 


yields 

dr = d, (90) 

a = ^ sin d, (91) 

/S = (k'- — k- sin2 d)^ ~ k', (92) 

and using these results in equations (88) and 
(89), one has 

(1 + r)k sin d = (SA, (93) 

(1 — r)k cos d = aA. (94) 


Finally, eliminating A from these two equations, 
one has 

. 2a tan d . 2k sin d tan d 
7- 1- — ^ 1-p-, (95) 

I r 17 ~ 1 — I ) sin d tan 6 , (96) 







44 


BASIC THEORIES 


The dimensionless quantity {2\i'o)/QC-)^- is of 
the order 7 X foi" castor oil at 10 kc and 
20 C, and thus we see that the reflected energy 
differs from the incident by less than 1 per cent 
at 45° incidence; even at 85°, the dissipation is 
only about 7 per cent of the incident energy. 
The approximate formula, equation (96), is 
valid until the second term on the right becomes 
of order 1, and we can see that this occurs only 
when the incident wave is within a degree or 
so of grazing incidence. 

The reflection-conversion mechanism was at 
one time regarded by the writer as a possible 
cause of inefficiency in transducers. If the match 
to the water is so poor that the equilibrium 
energy density of the standing-wave pattern 
inside the transducer is large compared to that 
in the water just in front of the diaphragm, 
this mechanism might cause appreciable dissi¬ 
pation. In a well constructed transducer, how¬ 
ever, the foregoing result makes it appear ex¬ 
tremely unlikely that this mechanism is of any 
practical importance. 


22 DIELECTRICS 

Just as in the last section we discussed pure 
elasticity, in this section we will discuss the 
theory of pure dielectrics, reserving the cou¬ 
pling of such systems to the next section. 

The theory of linear dielectrics is complicated, 
being analogous to that of diamagnetism. Cer¬ 
tain cuts of RS are both nonlinear and strongly 
temperature dependent in the range of sea tem¬ 
peratures, and hence even more complicated. 
The practical use of such cuts is limited almost 
entirely to extremely small receivers, where it 
is important to get as large a capacitance as 
possible so that the capacitance between the 
two conductors will not shunt the signal too 
severely. No satisfactory theoretical treatment 
of such systems is known, and we shall there¬ 
fore confine this discussion to dielectrics in 
which the polarization is a linear function of 
the components of the total electric field. This 
approximation is entirely adequate for Y-cut 
RS and all cuts of ADP, and therefore covers 
the great majority of all cases of practical 
interest. 


Dipole Distribution 

In this section we consider a continuous dis¬ 
tribution of dipoles without specifying what 
causes these dipoles; for example, they might 
be the dipole distribution of an electret. 

A dipole is the combination of equal positive 
and negative charges separated by a very small 
distance. The dipole moment, say m, is a vector 
which points from the negative to the positive 
charge and its magnitude is the product of the 
positive charge by the separation of the two 
charges. If such a dipole is placed at a point ri 
the potential at the point ro is given by 

m = m • gradi;i^, (97) 

ri2 M2 

in which rio = r^ — ri, and ri 2 is its magnitude. 
A continuous distribution of dipoles of volume 
density P therefore produces a potential given 
by 

(2) = f rfVP(l) - grad, j- 
p J n2 

We must now characterize the P distribution 
a little more precisely. First, we assume that P 
goes to zero at infinity fast enough so that all 
integrals, involved in the entire discussion, con¬ 
verge. Second, we assume that P is continuous 
and each of its components differentiable 
throughout all space, except for certain surfaces 
(closed or open) across which P suffers finite 
discontinuities in magnitude, direction, or both; 
these surfaces are the boundaries between dif¬ 
ferent materials in the system, crystal to air, 
crystal to electrode, etc. These conditions will 
be satisfied in any real physical problem. 

The first term in the right member of equa¬ 
tion (98) may be transformed to an integral 
over these surfaces. We divide all space into 
regions, as indicated in Figure 2, by auxiliary 
surfaces at all of whose points P is continuous. 
We then have two types of regions: type Ri, 
bounded wholly by surfaces across which P is 
continuous (including the surface at infinity, 
where P is zero) and type R 2 , partially bounded 
by a surface of discontinuity. These latter al- 






DIELECTRICS 


45 


ways occur in pairs, Ro and R 2 , with the discon¬ 
tinuous surface as a common boundary. 

Now the first term in the right member in 
equation (98) is the sum of the integrals over 
all these regions, which together fill all space. 




Figure 2. Auxiliary surfaces enclosing sur¬ 
faces of discontinuity. 

Using Gauss’s flux-divergence theorem, this 
sum is the sum of the flux of P/ri 2 out of each 
of these regions. The auxiliary surfaces always 
divide an Ri- and an i^o-type region and the con¬ 
tribution from them is therefore zero because P 
is continuous and the flux out of every element 
dS of an Ri is the negative of the flux out of the 
adjoining Ro. 

The integral is therefore just the total flux 
into the surfaces of discontinuity or, for better 
analogy with a result to be obtained for the 
second term, the negative of the total flux out 
of these surfaces. It is convenient to have a 
short name for the flux per unit area of a vector 
out of a surface, and we shall call it the surface 
divergence, 

surf div F = n • F + n' • F', (99) 

surf div = xp surf div F ... 

(if lA is continuous), ^ 

in which n and n' are the two oppositely directed 
unit normals pointing out of the surface on its 
two sides and F and F' are the values of the 
vector on the two sides, differing at most by a 
finite vector. If the normal component of the 
vector is continuous, its surface divergence is 
zero. It should be noticed that the symmetry of 
equation (99) avoids all ambiguity as to signs. 
The potential, equation (98), now becomes 

(t)p(2) = — f dS surf divi 

' f'l2 

^^,f diV:P(l) ]. (101) 

ri2 

This is the potential that would be produced in 


vacuum by a surface and volume distribution 
of electric charges of density 

(Tp(l) = - surf divi P(l), (102) 

pp(l) = - divi P(l). (103) 

However, this charge distribution must not be 
confused with a free charge distribution that 
can move about in the material. It is the so- 
called bound charge and arises solely from lack 
of cancellation between neighboring dipoles. It 
is interesting to notice that the total of all this 
charge is zero, 

Q-p = — J" surf div PdS — div Pc^V = 0, (104) 

because, upon transforming the second integral 
into one over the surfaces of discontinuity, just 
as we transformed the exact divergence in equa¬ 
tion (98), it exactly cancels the first term. 

We must now examine the general properties 
of (pp. First, it is everywhere continuous, be¬ 
cause it can be regarded as the potential of the 
distribution given by equations (102) and (103) 
and only a doublet (or higher) layer can give 
a discontinuity to an electrostatic potential. 
Second, at every point not on a surface of dis¬ 
continuity, it must satisfy Poisson’s equation 

V^4>p = — 47rpp = Ltt div P. (105) 

Finally, the normal component of its negative 
gradient Ep suffers a discontinuity at the sur¬ 
faces of discontinuity given by 

surf div Ep = 4x(rp = — 4x surf div P. (106) 
Introducing the electric displacement defined 
by 

Dp = Ep + 4xP, (107) 

all these relations are summarized by the simple 
results 

div Dp = 0, (108) 

surf div Dp = 0, (109) 

(pP (continuous everywhere), (HO) 

in which the last, equation (110), includes the 
more commonly used statement that the tan¬ 
gential component of E is continuous and is, in 
fact, more stringent. 

In conclusion, it should be emphasized that 
we have so far not developed a theory of dielec- 








46 


BASIC THEORIES 


tries, because we have said nothing about the 
cause of P. If the dipole distribution is given 
(e.g., a given electret which is not further 
polarizable), then equations (105) and (106) 
constitute a generalized Dirichlet boundary- 
value problem: a field equation with given non- 
homogeneous term, together with given values 
of the surface divergence of the field on certain 
surfaces. 

We are not interested here in this case and it 
is mentioned only for contrast and to emphasize 
that we are at liberty to postulate any pheno¬ 
menological relation between P and other quan¬ 
tities without invalidating our present results. 
Two different postulates will in fact be made: 
one in the next section to get the linear theory 
of pure dielectrics, and a more general one in 
the following section as a step toward the linear 
theory of piezoelectricity. 

^ Linear Dielectric 

To get the theory of linear dielectrics, we now 
need to make just two additions to the previous 
section: to include the possibility of a volume 
and surface distribution of free charges, and to 
assume a relation between the polarization and 
the total electric field at any point. The potential 
of a volume and surface distribution of free 
charges is given by 

+ ( 111 ) 

and by the same methods of the previous sec¬ 
tion, we obtain 


D, = E„ 

(112) 

div D/ == 47rp/, 

(113) 

surf div D/ = Airaf, 

(114) 

(f)f (continuous everywhere). 

(115) 

The potential of a superposition of dipoles 
and a volume and space distribution of free 
charges therefore satisfies 

(j) = 0 ^, -|- (continuouseverywhere), (116) 

E = E/- -)- E; = — grad 4>, 

(117) 

D = D/j -)- D/ = E 47rP, 

(118) 

div D = 47rp/, 

(119) 

surf div D = 47rcr/. 

(120) 


We now assume that the polarization is caused 
by the total field (7iot the field of the free 
charges only, because there can be no way of 
distinguishing between the two parts of the field 
by observations at a point) and we therefore 
have some kind of a phenomenological equation 
of state, 

P = P(E,). (121) 

The most complicated form of the general equa¬ 
tion (121) which is susceptible to treatment, 
and fortunately one which gives a good descrip¬ 
tion of a majority of dielectrics, is 

P, = Xr.E, or Dr = KrsE„ (122) 

= Srs + 47rXrs, (123) 

in which the susceptibility and dielectric tensors 
and which will later be shown to be 
symmetric, are otherwise arbitrary so far as 
the theory is concerned. They must therefore be 
evaluated and shown by experiment to be con¬ 
stants for many substances (in any one frame). 

Inserting equation (122) into equations 
(116) to (120) now gives a completely deter¬ 
minate Dirichlet problem if and are given. 
In our problems, is zero everywhere and is 
zero except on electrodes, and there its value 
is given (zero) only for an open-circuit receiver. 
Equation (120) must therefore be used to cal¬ 
culate o,, and we adjoin the boundary condition 
(f) constant over all electrodes (given for a trans¬ 
mitter, to be calculated for a receiver). How¬ 
ever, we are not yet ready to show how approxi¬ 
mate solutions of these problems are found; we 
do not yet have the theory of piezoelectric prob¬ 
lems, because we have as yet established no 
coupling between the elastic (see Section 2.2.1) 
and dielectric systems. 

We now calculate the potential energy of the 
distribution of free charges and the induced 
polarization. Let us imagine that the free 
charges in a distribution, both volume and sur¬ 
face, to be increased slightly. The work done, 

f Mason neglects the electric depolarizing field Ep in 
treating crystal plates, together with a further de¬ 
polarizing field of piezoelectric origin. We shall see 
later that in his approximation, this is justifiable; how¬ 
ever, in higher approximation and even in first 
approximation for other shapes, the depolarizing fields 
should be included.^''. 12 







PIEZOELECTRICS 


47 


which is equal to the increase in potential 
energy, is given by 


511 — J (f)8crj<iS J 4>8pjdV. (124) 

It should be noticed that we do not include 
terms for the increment of the dipole distribu¬ 
tion, because this increment is not independent 
but is determined by the increment to the free 
charges. The only way that an external system 
can do work is by bringing up further free 
charges, since to interfere with the polarization 
would be to violate equation (122). 

In equation (124), we express the increments 
to the distributions in terms of D, using equa¬ 
tions (119) and (120), and then transform the 
surface integral to one over the volume. 


4x5ir ^ J*( ~ ^ b\y)dV 

-f 


E • 5D(/y. 


(125) 


This result is so far perfectly general, being 
independent of the special linear relation, equa¬ 
tion (122). However, to integrate it, we must 
take advantage of this relation, obtaining 


E • T>dV (126) 

This total work done by the mechanical forces 
which brought up the free charges is just the 
potential energy of the system. It should be 
noticed that it is the final distribution of these 
mechanical forces which holds the charges in 
position. 

The localization of the energy in an electro¬ 
static field has been the subject of considerable 
debate, but at least we see that we will get all 
the potential energy if we assume that it is 
distributed throughout space with a density 
E^.K,.^EySz. We notice that this is a quadratic- 
form in the components of E and can reach two 
conclusions from this fact: no system can be in 
equilibrium under the action of electrostatic 
forces alone (Earnshaw’s theorem) and, hence, 
all the principal values of must be positive; 
also, if had an antisymmetric part, it would 
contribute nothing to the potential energy but 
would contribute a component to D which would 
be perpendicular to E, as shown by equation 



(122), a component against which no work is 
done since it falls out of the potential energy; 
hence, must be symmetric and, with it, 
Having separately developed the theory of 
linear elastic and dielectric systems, we are now 
ready to couple them and thus obtain the theory 
of linear piezoelectricity. 


PIEZOELECTRICS 

A piezoelectric system is an elastic and a 
dielectric system, both nonisotropic, occupying 
the same region. The essential feature that 
makes such a system so interesting and useful 
is that the two component systems do not func¬ 
tion independently, but are coupled so that me¬ 
chanical forces produce electric polarization 
and electric fields produce elastic deformations. 
Without this coupling, it would be impossible 
to convert electric energy to mechanical (trans¬ 
mitter) or mechanical to electric (receiver) ; 
with it, we have an electromechanical trans¬ 
ducer. 

This coupling can be included in the theory 
by taking over all of the fundamental relations 
of pure elasticity and pure dielectrics, but alter¬ 
ing the elastic and dielectric equations of state 
and making corresponding alterations in the 
energy density, propagation equations, and 
boundary conditions. The strain and the stress 
tensors each have six distinct components, and 
the pure elastic (linear) equation of state, 
equation (25) of Section 2.1.4, furnishes just 
six linear relations between them, so that if 
either tensor is known the other may be calcu¬ 
lated. Similarly, the electric displacement and 
the electric field each have three components 
and the pure (linear) dielectric equation of 
state, equation (122) of Section 2.2.2, supplies 
three linear relations between them, so that if 
either vector is known, the other may be calcu¬ 
lated. Regarded together, there are eighteen 
distinct quantities between which nine equa¬ 
tions of state have been assumed. In the absence 
of coupling, these nine relations fall apart into 
two sets: the six elastic and the three dielectric 
equations. This must be altered so that we still 
have nine relations between eighteen quantities, 
but they do not fall apart. It is a matter of con- 




48 


BASIC THEORIES 


venience how we choose these relations; any 
nine quantities may be expressed in terms of 
the other nine, and the form used will depend 
on the particular application. New material con¬ 
stants will need to be introduced, correspond¬ 
ing to the piezoelectric coupling, and it will be 
shown that there are at most eighteen of these 
new constants, this number being reduced by 
any symmetry. 

It should be emphasized that the potential is 
an internal field quantity^® and that the relation 
between it and the voltage applied to the elec¬ 
trodes is established through the boundary con¬ 
ditions. This is a necessary variation from 
Mason’s treatment since regarding the electric 
field as an externally given quantity corresponds 
to neglecting both the electric and mechanical 
depolarizing field and, although this is valid in 
lowest approximation and for certain types of 
geometry, it is not a generally valid approxi¬ 
mation. Physically, this corresponds to the fact 
that the polarization at a point is caused by 
the field at that point, it being immaterial that 
part of this field is caused by free charge and 
part by polarization. One consequence of this 
variation is that the proper energy density of 
the vacuum field E-/S 71 will be included in the 
energy of the system, while Mason’s expression 
for the total energy density vanishes if both 
the strain and the polarization vanish. We there¬ 
fore have a problem involving four field quan¬ 
tities, the three components of the displacement 
and the electric potential 4>. 


Energy Density and Equations 
of State and of Propagation 

In Section 2.1.4, we obtained the potential 
energy density of a linear elastic system and 
derived the elastic equations of state by equat¬ 
ing the work done, in an infinitesimal additional 
displacement, to the increase in total potential 
energy. Similarly, in Section 2.2.2, we calculated 
the potential energy density of a pure dielectric. 
We must now apply this same treatment to the 
piezoelectric case. 

If the system is initially in static equilibrium 
under the combined action of external body and 
surface forces, together with the forces neces¬ 


sary to hold any free charges in place, the work 
done in an infinitesimal displacement is 

Jdv(^Pr.is„ + ^^^. (127) 


The potential energy density must vanish if 
both the strain and the electric displacement 
vanish, and otherwise must be positive. Again, 
the simplest function satisfying these condi¬ 
tions is a quadratic form, and neglecting all 
higher-order terms, we have 


Txr /IN. _ - , DrKrs~^D, 

Ve \2j^PQrs^pq^rs “t“ q 

OTT 

I Dr/ rpqS p q 

47r ’ 


(128) 


in which the notation^ has been chosen so as to 
agree with Mason’s.The first two terms are 
the pure elastic and the pure dielectric energy, 
the first for an electrically shielded crystal 
(D = 0), and the second for a mechanically 
blocked crystal = 0), while the third term 
is the coupling energy. The new coefficients 
form a third-rank tensor,^ symmetric in the last 
two indices and therefore having at most eight¬ 
een distinct components; if this tensor is zero, 
the systems separate and there is no electro¬ 
mechanical energy conversion; if small, high 
voltage is required to convert electric to me¬ 
chanical energy. 

The increase in potential energy correspond¬ 
ing to a variation of the strain and electric dis¬ 
placement is 




Equating this to the work done, equation 
(127), and taking account of the arbitrariness 

R Compare equation (C.43) of reference 12 with 
equation (130) below. The above energy density is just 
E-/8k greater than Mason’s equation (C.30).i2 This 
difference, the self-energy of the field, corresponds to 
taking the field into the system instead of regarding it 
as externally given. 

See Section 2.3.3, where the connection between the 
tensor and matrix representation of all of the ma¬ 
terial constants, and their transformation properties, 
are discussed. 









PIEZOELECTRICS 


49 


of the region R, we get for the generalized equa¬ 
tion of state 


P 


pq 


Er 


dW 

dSpq 

AivdW 

dDr 


C D 


+ 


f rpqSp 


Drfrpq 

47r ’ (130) 

+ Krs-^ D,. 


An alternate form of equation (130), more 
suitable for getting the field equations, is ob¬ 
tained by solving the second for D, and substi¬ 
tuting into the first 


P pq CpqrsSrj 4“ f rpqEri 

Dr = — 4:TTfrpqSpq + KrsEs, 

f mpq K rttn f nrs 


f rpq — 


C pq rs 
Krm fr, 


4ir 


47r 


(131) 


(132) 


Now in obtaining the equation of motion of 
an element of volume, equation (21) of Section 
2.1.3, we had no occasion to inquire into the 
cause of the stress. It was sufficient that the 
force between neighboring parts of the system 
could be described by a stress tensor. Similarly, 
in obtaining the result that lines of induction 
begin only on free charges, equations (119) 
and (120) of Section 2.2.2, it was sufficient that 
the field be produced by a distribution of free 
charges and dipoles regardless of the origin of 
these distributions. Therefore, these relations 
are still valid provided the coupled equation of 
state, equation (131), is used; and the coupled 
propagation equation is therefore 


piip — CpqrsUr.s.q f rpq <t>,r,q 4“ Bp, /I OO j 

A-irfrpq Up,q.r 4 “ Krs (j^.s.r = “ ^TTPj, 

in which the change of to „ is possible be¬ 
cause of the symmetry properties of the tensors 
c and /. These are four field equations govern¬ 
ing the propagation of the four field quantities 
2 ^^ and (f), if the body force and charge density 
are given. We have retained these quantities, 
even though they are zero in all our problems, 
because equation (133) may also be regarded 
as the explicit solution of the problem of what 
body force and charge density are necessary to 
support an arbitrarily assumed displacement 
and potential. This enables us to estimate the 
error in an approximate solution. 

If we set / to zero, equation (133) becomes 


the propagation equation in pure elasticity and 
Poisson’s equation in pure dielectrics, and the 
coupled system separates into two independent 
systems. If the coupling is made unilateral by 
setting f to zero in the second equation, we 
have the approximation in which the field is re¬ 
garded as externally given and only the me¬ 
chanical depolarizing field is neglected. The re¬ 
sultant system will not obey the reciprocity 
principle. 


2.3.2 Linear Dissipative Piezoelectric 

Boundary-Value Problem 

Surface Dissipation 

We are now in a position to condense the fore¬ 
going results by formulating a boundary-value 
problem governing the behavior of a quite gen¬ 
eral linear dissipative piezoelectric system. 

The time-dependent boundary-value problem 
is readily formulated, but its solution in any 
other case than steady state appears so hopeless 
that it is of doubtful practical value; accord¬ 
ingly, it is assumed that all quantities have time 
variations like exp ({coi), and only the amplitude 
problem is stated. However, transient problems 
can then be successfully attacked by the well- 
known Fourier integral method. 

The fundamental field quantities are taken to 
be three displacement components Upix,y,z,t) 
and the electric potential <t>{x,y,z,t) ; this choice 
avoids adjoining differential identities which 
would be necessary if polarization, field, or elec¬ 
tric displacement were used. It is permissible to 
ignore the magnetic field caused by variations 
in the electric field because, even at the highest 
frequency in practical use, the magnetic field so 
created is extremely small and the resultant in¬ 
ertia and dissipation introduced is negligible. 

From these four fundamental field quantities, 
the stress and electric displacement, auxiliary 
field quantities, are derived according to 

P pq — CpqrsUr.s f rpq4>,r ■> (134) 

Dp = ^Trf prsUr,3 Kpq(t>^q. (135) 

The use of the asymmetric derivative tensor 
instead of the symmetric strain tensor is 








50 


BASIC THEORIES 


permissible since the symmetry of and 

/p,,, selects only the symmetric part anyway. 

The propagation equations, in steady-state 
form, are 

pOrUp -|- P pri,q + Bp = 0, (136) 

div D = Dp,p = 4xp/. (137) 

These are four field equations for determining 
the four field quantities and (f>, and they re¬ 
quire only boundary conditions to form a de¬ 
terminate problem. It is scarcely necessary to 
remark that all quantities are now complex am¬ 
plitudes depending on spatial variables only. 

The body force and the free charge density 
are assumed to be zero in all problems treated 
in this volume, except that in the next section a 
complex dielectric constant is introduced, which 
corresponds to assuming a volume conductivity 
in insulating materials. These quantities are re¬ 
tained here because the propagation equations 
explicitly give the extraneous body force and 
free-charge density corresponding to an as¬ 
sumed displacement and potential, and hence 
enable the error in an approximate solution to 
be estimated. 

Before the boundary conditions can be formu¬ 
lated, it is necessary to define the region. In 
doing this, it should be recalled that a piezo¬ 
electric system is actually a superposition of 
two systems, one elastic and the other dielectric, 
with coupling between the two. Hence the 
boundary-value problem contains those for pure 
elastic and pure dielectric systems as special 
cases provided we allow for discontinuous 
changes in all the material constants. For ex¬ 
ample, in a crystal transducer, f falls suddenly 
to zero across crystal surfaces; likewise the c 
and K tensors undergo discontinuous changes. 
Furthermore, it may be advantageous to ex¬ 
clude certain interior regions, occupied by Cor- 
prene, etc., from the elastic domain, regarding 
them as energy sinks represented by a complex 
impedance over the bounding surface of the ex¬ 
cluded region; however, it would be quite wrong 
to exclude these from the domain of the electric 
potential since some of these materials exhibit 
high dielectric dissipation. 

We are therefore led to the region shown in 
Figure 3. The total region R is bounded ex¬ 
ternally by a conducting sheath and is held at 


zero potential. It may extend to infinity; if not, 
it is regarded as terminated by a continuous 
distribution of mechanical generators exerting 
a force per unit area of amount F and having in¬ 
ternal impedance characterized by a specific 
acoustic stiffance' tensor 5^^, both of which may 
be functions of position on the surface and of 
frequency. The elastic region is R with the im¬ 
pedance regions excluded; the interactions 
across these surfaces are characterized by a 
stiffance tensor and, if any are active, by an F 
just as with the external boundary; the elastic 
region will be designated as 7? — Z. 

The metal conductors are closed regions as¬ 
sumed to have infinite conductivity. The poten¬ 
tial is therefore constant within them, and the 
potential domain is therefore R — c; however, 
the interior of these conductors is a part of the 
elastic domain. 

Finally, the whole region R will usually be 
separated into subregions in which the material 
is homogeneous (but not necessarily isotropic) ; 
a surface of discontinuity may correspond to a 
discontinuity in any of the material tensors 
f or K. 

The boundary conditions to be adjoined to the 
above propagation equations, to form the com¬ 
plete boundary-value problem, are therefore 


P pqTlq Fp S pqliq 

on impedance and external surfaces, ^ 


(Ppg - Ppq)nq = 0 

on all surfaces of discontinuity, 
n pD p = 4x0"/ 

n outward, on conductors, 

{Dp - Dp')np = 0 (141) 

on nonconducting surfaces of discontinuity, 

(j), Urn 

continuous everywhere, 

0 = F, 

constants on and inside conductors. 


(139) 


(140) 


(142) 


(143) 


Internal Viscous and Dielectric 
Dissipation 

The foregoing steady-state boundary-value 
problem includes the possibility of radiation or 


i The representation of inert loads, both tangential 
and normal, by a stiffance tensor is discussed in 
reference 4, 






PIEZOELECTRICS 


51 


true dissipation into impedance surfaces, but 
the interior of each medium has been regarded 
as conservative. This is a good approximation 
for crystals, steel, etc., since the internal dissi¬ 
pation in these materials is negligible compared 
to the radiation losses for a crystal working 
into water. 

However, there are dissipation processes, 
known to be of practical importance, which 
cannot be represented by an impedance surface. 
These include dielectric losses, which may be 
serious in Corprene, and internal elastic dissi¬ 
pation in Corprene, rubber, etc. Accordingly, it 
seems advantageous to broaden the general 
theory to include these types of dissipation. 

Dielectric dissipation can be included, in the 
steady state, by adding an imaginary term. 



to the dielectric tensor, in which is the con¬ 
ductivity tensor occurring in Ohm’s law. This 
tensor is assumed to be symmetric, since any 
antisymmetric part would cause a current to 
flow without any corresponding dissipation. It 
is also assumed to be positive-definite, since 
otherwise certain directions of the field would 
correspond to negative dissipation. 

Internal elastic dissipation can be included 
by adding to the stress a term linear in the 
strain velocity, an obvious generalization of the 
Stokes-Navier equation. In the steady state this 
becomes 

^J■!)qrs Srs “ i<J^^^pqrs U r ,s- 

The viscosity tensor is symmetric in the 
first two indices because otherwise the well- 
known requirement of symmetry of the stress 
tensor would be violated. It is symmetric in the 
second pair, because the strain is. Finally, un¬ 
less it is symmetric with respect to exchange of 
the first and second pair of indices, there will 
be velocity dependent forces which, like the 
magnetic deflection of a charge, cause no dissi¬ 
pation ; such forces are not considered here. Ac¬ 
cordingly, the viscosity tensor is assumed to 
have the same symmetry properties as the 
elastic stiffness tensor. 

Thus by allowing and to become com¬ 
plex, but maintaining the same symmetry, the 


possibility of internal viscous and dielectric 
losses is included in the foregoing boundary- 
value problem without changing its formal 
structure. 

In the next section, this boundary-value prob¬ 
lem, complete with boundary conditions as well 
as propagation equations, is shown to be deriv¬ 
able from a variation principle. Solutions are 
then obtained by the semidirect method, for the 
case of a rectangular crystal plate, first in the 
Mason approximation and then in a higher ap¬ 
proximation which includes additional effects of 
practical importance. 


2.3.3 Formulation; Transformation 

Under Rigid Rotation 

In formulating the general theory of elastic, 
dielectric, and piezoelectric systems, we are led 
inevitably to the tensor analysis because certain 
sets of physical quantities (e.g., the components 
of the strain, the stress, the electric displace¬ 
ment, the elastic, dielectric, and piezoelectric 
constants) demand consideration as a whole 
just as, in a special case of this, a force de¬ 
mands consideration as a single physical in¬ 
fluence even though we need three quantities to 
represent it. 

There are several great, interrelated advan¬ 
tages to the tensor formulation i-" (1) the above 
simplification of concepts whereby, for example, 
we can think about the stress, instead of think¬ 
ing of six quantities; (2) its treatment of all 
coordinate frames on an equal basis, thus free¬ 
ing us from preoccupation with geometric de¬ 
tails of a particular frame in general problems 
in which these details are irrelevant; (3) the 
formal simplicity of the transformation of a 
representation in one frame to that in another; 
(4) the great economy in notation which often 
enables the results of the lifework of great 
physicists to be developed and written down in 
a few pages, etc. 

j It is unfortunate that tensor analysis was not 
sufficiently developed to be more useful to \V. Voigt^ in 
his great work on crystal physics. All through his long 
book (about 800 pages), one feels that he is searching 
for refinements to this valuable tool, refinements which 
came shortly afterwards through the work of Levi- 
Civita and others. 





52 


BASIC THEORIES 


These advantages disappear to some extent 
when we attempt a specific calculation: a co¬ 
ordinate frame, having geometry as simply re¬ 
lated to the system as possible, is chosen; it is 
inconvenient to make numerical arrays of the 
elastic constants, a fourth-rank tensor, etc. For 
these and other reasons, there are many advan¬ 
tages to the commonly used matrix formulation'" 
for specific applications. We must therefore de¬ 
velop this formulation and its connection with 
the tensor formulation, here restricting our¬ 
selves, however, to Cartesian coordinates. 

The matrices involved are rectangular rather 
than square, several having only one row or one 
column. Taking advantage of the symmetry, 
there are at most only six distinct elements in 
the strain or stress tensor, twenty-one elastic, 
and eighteen piezoelectric constants, etc. These 
various quantities can be conveniently arrayed 
by using matrices which have at most six rows 
and columns, and we therefore introduce new 
indices a,h,c, . . . taken from the early part of 
the alphabet and always ranging from 1 to 6 . 

The stress tensor now becomes a matrix of 
six rows and one column (to avoid confusion 
with the polarization, it will be represented by 
X in this section). 


X„. 


a 

Pu 1 

P 22 2 

P 33 3 (144) 

P 23 4 

P31 5 

P12 6 


We shall also have occasion to write the trans¬ 
posed matrix. 


0 1 2 3 4 5 6 

X _ p p n p n p (J-40) 

•a — 11 22 -1 33 23 31 12 

The dot shows which index is absent and in 
most cases we can drop it. When writing these 
matrices without their indices, they become 
simply X and X^ in which the special super¬ 
script is not a running index but merely means 
“transposed.” 

In a similar way, the strains are arrayed as 


The matrix formulation is developed by Bond.i3 
However, the connection between the tensor and matrix 
formulation is not developed. 


one column or one row matrices. The transposed 
strain matrix is 


s.a = Sii S 22 S 33 2s23 2s3i 2si2 = s', (146) 

with a corresponding form for s itself. It should 
be noticed that the off-diagonal elements of the 
strain tensor are doubled to form the corre¬ 
sponding elements of the matrix. 

It is convenient to have a condensed notation 
for showing the connection between the various 
tensors and the corresponding matrices. This 
can be done by introducing connection matrices, 
which are quantities having mixed indices. 


X.. = Ca 


Sa- — D apqS p 


C apq ^ ap^ pq ^ 2 )Pia—^)P<l 

(not summed on p), 

D apq ~ ^ap^pq P{a— 3 )pq 

(not summed on p), 


(147) 

(148) 


in which is a permutation matrix, 1 if mpg 
is any permutation of 123 and 0 otherwise, or 


Pmpq = NmpNp,N,„r, Mrs = (1 " 5,,). 
The reciprocal connections, whereby the mat¬ 
rices are converted to tensors, are easily shown 
to be 


P pq — X(,D apq, ^ pq SqCapq, ( 149 ) 

CapqDars = hi^pr^qs + ^ps^qr), ( 150 ) 

SO that C and D are reciprocally related. 

Substituting these matrices into the energy 
density and equations of state, equations (128) 
and (130) of Section 2.3.1, they become 


W = ^aCabSb + 


DrKrs-^Ds ^ DrfraSg 


= Is'cs -f 


DK D , Dfs 


Stt 


+ 


4r 


X = cs + ^^, 


(151) 


(152) 


E =fs 

= Ca 


fX H, 

iCijrs Cft, 


(153) 


Oab ^apqOpqrs^ 
fra f r pqC apq‘ 

Thus, the elastic tensor becomes a 6 by 6 
symmetric matrix, and the piezoelectric tensor 
/,.p^ becomes a matrix of three rows and six 
columns. The dielectric tensor remains a 3 by 3 
symmetric matrix, and the vectors E and D be¬ 
come matrices of three rows and one column. 








PIEZOELECTRICS 


53 


It is extremely easy to manipulate these 
matrix equations, the only necessary precau¬ 
tion being to preserve the order of factors in 
each term, since multiplication is not necessar¬ 
ily commutative. For example, to obtain the 



Figure 3. The domain in which the linear 
dissipative piezoelectric boundary-value problem 
is defined. 


analogues of equations (131) and (132) of Sec¬ 
tion 2.3.1, we need only premultiply the second 
of equation (152) above by K, solve for D, and 
substitute into the first; 


X = cs +lE, 

D = — 47r/s + KE, 


t 



(154) 


(155) 


We can now obtain the rotation transforms 
of the material constants in a form requiring, 
at most, the multiplication of 6x6 matrices, and 
therefore suitable for numerical calculations.^ 
We first notice that the energy density, equation 
(150), separates into three terms, the first two 
of which we interpret as the proper energy of 
the elastic and dielectric systems, respectively, 
and the third as the coupling energy between 
these systems. This separation must be possible 
in every frame and hence each of these terms is 
a tensor of zero rank. Since the strain and dis¬ 
placement are matrices derived from tensors, 
the transforms of the material constants are 
uniquely determined by the transformation 
properties of these matrices. 


1 These relations are given by Bondd^ but there are 
some errors in this otherwise very valuable paper. 


If the frame is rigidly rotated, the new co¬ 
ordinates are 

Xp = apqXq or x' = ax, (156) 
in which a is orthogonal, The trans¬ 

formation of the vectors E, D, and P follow the 
same law. 

The transformation of the strain tensor, in 
the opposite direction to allow the old quantities 
in the energy density to be expressed in terms 
of the new, are 


Spg arpttsqS rsj (15T) 

and by replacing the tensors by the correspond¬ 
ing matrices, we find 

s„ = or s = Ts', 

rp _ 7 ^ n n n 

ab ■^dpfj^rpO'sq^ ars* 

The matrix T has been defined as above so as to 
be identical to Bond’s alpha.^^^ 

We now replace the old strain and electric 
displacement by the new, the former according 
to equation (158) and the latter according to 
equation (156) after inversion, D = a^D'. The 
three terms in the energy density then become 

s'cs = s'‘c's'; c' = TcT . (159) 

DK-^D = D'X'-^D'; K'-^ = aK-^a‘. (160) 

Dfs = /' = afT. (161) 

These are explicit formulas for calculating the 
material constants in a rotated frame from 
those in the original frame. Although these cal¬ 
culations are bound to be laborious, the matrix 
formalism enormously shortens them, in fact, 
to such a point that a calculation that is other¬ 
wise not practically feasible, can be carried 
through in an hour or two. The steps in this 
calculation are as follows: Write out a and 
calculate the elements of the 6x6 matrix 
then carry out the matrix multiplications indi¬ 
cated in equations (159), (160) and (161). 
These multiplications are made more easily, 
and the probability of mistakes reduced by rul¬ 
ing grids 6x6, 6x3, and 3x3 with quite large 
squares and then cutting the paper with no 
margins so that a prefactor may be fitted di¬ 
rectly to a postfactor. 

These transformations are of value even 











54 


BASIC THEORIES 


when the numerical value of the constants are 
not known, because they enable us to calculate 
the proper orientation to make certain con¬ 
stants zero (e.g., so that a field normal to a 
plate will cause no shear stress) merely from 
a knowledge of the symmetry class of the 
crystal. 

Symmetry Reduction of the Matrices 

Nearly all crystals possess some symmetry, 
that is, by performing certain rotations and/or 
reflections, the resultant is indistinguishable 
from the original, provided we ignore its finite 
size. Each symmetry class is characterized by 
some finite subgroup of the rotation-reflection 
group, the extremes being the completely asym¬ 
metric crystal, which admits only the identity, 
and the completely symmetric (isotropic) sub¬ 
stance, which admits the entire continuous 
rotation-reflection group. 

Corresponding to each symmetry class, there 
will be certain relations existing between the 
material constants such that the higher the 
symmetry, the fewer independent parameters 
are needed to determine all the others. Crystals 
from which practical transducers can be made 
represent a nice balance between too little and 
too much symmetry. If too little, their field-in¬ 
duced deformations will be so complicated that 
nothing resembling a piston motion can be ob¬ 
tained; if too much, all their piezoelectric con¬ 
stants vanish. 

To determine the special structure of the 
matrices of a crystal of a known class, we need 
only transform the matrices in accordance with 
equations (159), (160), and (161) of Section 
2.3.3, using all the distinct transformations of 
its particular subgroup, and in each case de¬ 
mand that the matrix be unchanged. In this sec¬ 
tion, the resultant reduction of the matrices of 
RS and ADP are calculated. 

First, we must notice that while there are 
several different elastic matrices (c, the elastic 
matrix for zero electric displacement; c, for 
zero field, etc.), they all have the same sym¬ 
metry properties. The same remark applies to 
the various piezoelectric matrices (/, the strain- 
field matrix for zero electric displacement; /, 
the stress-field matrix for zero strain, etc.) 
and to the various K matrices {K, the clamped 


dielectric matrix for zero strain; K^, the free 
dielectric matrix for zero stress, etc.). This can 
be proved as follows. Any two tensors having 
the same rank also have the same symmetry 
since they will transform in the same way under 
the rotation group corresponding to the particu¬ 
lar symmetry of the crystal. The conditions 
from which the symmetry is determined will 
therefore be identical for both; the only ques¬ 
tion is therefore whether, in going over to the 
matrix representation, the same connection 
formulas are involved. This is true for c, c, etc.; 
/, f, etc.; K, K^, etc. To illustrate a case in which 
it is not necessarily true, let us solve equation 
(152) of Section 2.3.3 for the strain: 

s = c-'X - (162) 

47r 

The compliance matrix c~'^ can be expressed as 
a fourth-rank tensor just as c itself, and there¬ 
fore has the same symmetry properties as c 
when so expressed. However, the connection 
formula is 

Cab~'- = DapqCpgrs~^Di,rs (163) 

which differs from equation (153) of Section 
2.3.3 in using the D instead of the C connection 
matrix. Thus, while the matrices and 

both have the same symmetry properties, 

these properties are not necessarily the same as 
those of c and c. This difference goes back to 
the different way in which the strain and stress 
tensors are converted to matrices. If instead of 
putting a factor 2 on the off-diagonal elements 
of the strain tensor to get the last three ele¬ 
ments of the strain matrix, equation (146) of 
Section 2.3.3, a factor 2^ had been put on the 
off-diagonal elements of both the strain and 
stress tensor, then the C and D connection 
matrices would be identical and this difference 
would disappear. In any event, we can always 
calculate from c, having first determined 
the symmetry properties of c ; and, for the par¬ 
ticular symmetry of RS and ADP, c and c-^ 
actually have the same symmetry properties. 

Having established the principles of sym¬ 
metry reduction, we need only do detailed calcu¬ 
lations for c, /, and K, deducing all other 
matrices from these. We first consider RS, since 
the symmetry group of ADP includes that of 





PIEZOELECTRICS 


55 


RS, and we will thus simplify the calculations 
for ADP. RS belongs to the orthorhombic sys¬ 
tem, bisphenoidal (Bond’s class 6). Its axes are 
mutually perpendicular and 180° rotation about 
any one leaves the properties unchanged. Only 
two axes of the rotation need be considered, 
since the third is the resultant of any two. We 
therefore require that all matrices be unchanged 
by either of the two rotations 


1 

r 1 "1 

1 

r 1 

ttl = 

1 

1 

tH 

1 

_1 




The K matrix, transformed according to 
these two «’s, becomes {K and transform 
identically) 


left and right 3x3 matrices into which f may be 
broken for convenient treatment. For ag to be 
the same as g for both a’s, it is necessary that 
g = 0. Also, we see that h transforms just like 
K above, and hence must be diagonal. 

Turning now to c, we have 


c' = TcT 


r^ioi [piQi fiLo 1 
^0|a J L q\ rj L0|a* J 

P \Qa' 1 
_ aq\ara^ J’ 


in which p, q, and r are temporary notations for 
the 3x3 matrices into which c may be broken. 
Now we see that q transforms just like g above, 
and hence must be zero. Furthermore r trans¬ 
forms like h and hence must be diagonal. 


K[ — aiKa[ — P —h + 

L - + + 

r +-+ 

= a-^Kai = — 1 — 

L +- + 

For these diagonal a’s, the elements are not 
moved around but merely have their signs 
altered as indicated. Each of these transformed 
matrices must be identical with the untrans¬ 
formed, and hence all those elements which 
become negative upon transformation must 
actually vanish; hence we conclude that K is 
diagonal. Symmetry conditions can tell us noth¬ 
ing more about K, and actually all three of its 
elements are distinct for RS. 

Before transforming / or c, we must obtain 
the T’s corresponding to the two a’s. These are 
easily shown to be represented, for diagonal a’s, 
by the supermatrix 



T = 


I 0 
0 I a 


(166) 


The transforms of /, according to equation 
(161) of Section 2.3.3, are therefore 


r = afT^ = 


a|0 ~] r g\h 

Lo|oJ L ' 

ag\aha^ 


][^] 


0 


ha^-] 

0 J’ 


(167) 


in which a is to be given its two values Ui and 
ao, and g and h are temporary notations for the 



Figure 4. Symmetry-reduced matrix of RS. 


The c, /, and K matrices for RS, arrayed for 
convenience in a 9x9, are therefore as shown in 
Figure 4 (all elements not shown are zero). 

We now turn to ADP which has higher sym¬ 
metry, being a member of the tetragonal sys¬ 
tem, scalenohedral (Bond’s class 11). Like RS, 
its axes are mutually perpendicular and each is 
digonal, so that its matrices are reduced at 
least as much as those of RS. However, it has 
the additional symmetry of a tetragonal rota¬ 
tion-reflection axis (always taken as the z 
axis), such that rotation by 90° around this 
axis, together with reflection across a plane 
perpendicular to it {xy plane), leaves it un¬ 
altered. Only one of these 90° rotation-reflec¬ 
tions need be used, since all the others have 























56 


BASIC THEORIES 


been taken into account because of the three C 44 = c.-, 5 . To find the significance of ApA = p, 
digonal axes. The a and T matrices are we must do the multiplication, obtaining 


in which A is the matrix each of whose elements 
is the square of the corresponding element of a 
(not to be confused with the matrix a~). 

Transforming K, already diagonal because of 
the digonal axes, we find 

r K22 “I 

aKa‘ = Kn , (170) 

L iC33 J 

so that we conclude that Z 22 = 



Figure 5. Symmetry-reduced matrix of ADP. 


Transforming /, with the left 3x3 already 
zero and the right diagonal, we find that the A 
part of T does not enter, and hence we have 

h' = aha*, (171) 


n 0 1 0 n n Cn C 12 ci3 


ro 1 on 

ApA = 10 0 C 12 C 22 C 23 


1 0 0 

L 0 0 1J L ci3 C 23 C 33 _ 


0 0 1 _ 


(173) 

n C22 C12 C23 ~] 

C12 Cii Ci 3 , 

L C23 Ci3 C33 _J 


and hence we conclude that C 22 = Cn and 

^23 = C\3- 

The c, f, and K matrices for ADP, arrayed in 
a 9x9, are therefore as shown in Figure 5. 

The foregoing Figures 4 and 5 give all the 
independent parameters necessary to specify 
RS and ADP, but of course give no information 
as to the numerical values of these parameters. 
The constants are referred to the crystallo¬ 
graphic (mutually perpendicular) axes; in the 
next section, we will rotate these matrices so 
that they are suitable for studying 45° Y-cut 
RS and 45° Z-cut ADP. 


Matrices for Rotated Cuts 

The symmetry-reduced matrices of RS and 
ADP obtained in the previous section refer to 
the (mutually perpendicular) crystallographic 
axes. In this section we obtain these same 
matrices referred, however, to rotated axes. 
The rotations are 45° around the Y axis for RS 
and 45° around the Z axis for ADP. 

First consider ADP, since it is simpler. The 
a- and T matrices for a 45° rotation around the 
Z axis are 


a 



s 

c 


0 


0 

0 

1 


] 


which is the same as the transformation of K 
above and hence /14 = / 2 r.* 

Finally, the new c is given by 

e' = TcT = [A^] [A^] 

(172) 

^ ~ ApA\ 0 n 
0 \ara* J' 

Recalling that v is already diagonal, we see that 


1 if) 

0 0 cr 

1 if) 

0 0 — (7 

001 

0 0 0 

000 

c -s 0 

000 

s c 0 

— |cr \(y 0 

0 0 0 


in which c and s are abbreviations for the sine 
and cosine of 45° (or —45°) and a is the sign of 
s, -f 1 if the rotation is counterclockwise, —1 if 
clockwise. 

The detailed calculations involve a straight- 























PIEZOELECTRICS 


57 


forward application of equations (159), (160), 
and (161) of Section 2.3.3, but are a little long 
and hence only the results are given, arrayed 
in a 9x9 in Figure 6 (elements not shown are 
zero). 

The unprimed quantities in Figure 6 are 
those of Section 2.3.3 and refer to the crystallo¬ 


®‘ll '‘12 '13 


»»36 

''12 ®'ll '13 


-»^'36 

®I 3 ®I 3 '33 


0 


'44 

-Of,4 


C44 

of ,4 


'66 



af |4 

•<11 


.of,4 

*^11 

-<^»36 0 



Figure 6 . 

Matrix for 45° Z-cut ADP. 


graphic axes. The primed quantities, expressed 
in terms of the unprimed, are 


cii = 
C'n = 

cU = 


Cll ~l~ <^12 
2 

Cll + Cl2 
2 

Cll ~ Ci2 
2 


+ C66 


— C66 


(175) 


We now see that there is no difference be¬ 
tween +45° and —45° Z-cut ADP plates, if 
properly oriented, and that we may therefore 
take o = +1. If we rotate either plate 180° 
around its x or y axis (plate, not crystallo¬ 
graphic), it becomes indistinguishable from the 
other. This is exactly what is done, in polariz¬ 
ing crystals, if a small longitudinal compres¬ 
sion shows the wrong polarity. The identity of 
these two cuts is a consequence of the tetrag¬ 
onal rotation-reflection axis, and we must 
therefore be prepared for a more complicated 
situation in RS since it does not have this sym¬ 
metry. 


Turning now to RS, the a and T matrices for 
a 45° rotation around the Y axis are 


T = 


r c 0 -s “] 
a = 0 1 0 

L s 0 c J 


1 

6 

1 

2 

0 

1 

0 

1 

2 

0 

1 

2 

0 

0 

0 

— a 

0 

cr 

0 

0 

0 

0 

0 

0 

c 

0 

s' 

^cr 

0 


0 

0 

0 

“0 

0 

0 

—s 

0 

c 


(176) 


Applying these, in accordance with equations 
(159), (160) and (161) of Section 2.3.3, to the 
c, /, and K matrices arrayed in Figure 4, we 
obtain the corresponding matrices in the ro¬ 
tated frame. The results are given in Figure 7 
below, arrayed in a 9x9. The primed quantities, 
expressed in terms of the elements of the un¬ 
rotated matrices, are 


Cll 

Cl'o 

c'ls 

Cl's 

Cl's 

Cii 

Cie 

Css 

/u' 

/34 

K'n 

K'n 


Cll + C33 + 2 Ci; 

4 

C12 + C23 


+ Css, 


Cll + C 33 + 2ci3 

4 

Cll ~ C33 

4 ’ 

C12 ~ C23 
2 ’ 

C44 + Cee 
2 ’ 

Cee ~ C44 
2 ’ 

Cll + C33 — 2 ci3 

4 

/l4 ~ fz( , 

2 ’ 

fu + /se 
2 ’ 

7^11 + 7^33 

2 

Kn - 7^33 


(177) 


The matrices of Figures 6 and 7 are not con¬ 
venient for comparison of the properties of rec¬ 
tangular plates of 45° Z-cut ADP and 45° Y-cut 
RS, because the xyz frame is differently ori¬ 
ented with respect to the edges of the plates in 
the two cases. By rotating the frame, to which 



























58 


BASIC THEORIES 


Figure 7 is referred, ±90° around its present 
X axis, the new ;2: axis is normal to the plate, as 
in Figure 6 (it coincides with the crystal Y 
axis of RS), and the new x and y axes are 
parallel to the edges of the plate. The resultant 
matrix is given in Figure 8. 

We now see that the structure of the matrices 
of 45° Z-cut ADP is included in that of 45° 
Y-cut RS because the structure of Figure 6 is 
obtained from that of Figure 8 by setting the 



following elements to zero: (1) the 3 dilation- 
shear couplings; (2) the 2 off-diagonal shear- 
shear couplings; (3) the elements in the 14 and 
25 positions in the / matrix; (4) the off-diago¬ 
nal elements of K. 

Thus, one must expect that the motion of a 
rectangular plate of 45° Y-cut RS will be con¬ 
siderably more complicated than that of a 45° 
Z-cut ADP plate, and that this greater com¬ 
plexity will be aggravated by temperature de- 
pendances and nonlinearity; silicon carbide dust 
pictures verifying this expectation are shown in 
Figures 12 and 13 of Section 2.5.3. The Mason 
approximation neglects all motions except the 
simple longitudinal mode, and hence indicates 
only quantitative difference between RS and 
ADP; however, this difference appears very 
sharply as soon as a higher-order approxima¬ 
tion is made (see Section 2.5.3). 

In order to be able to treat rectangular plates 
of 45° Z-cut ADP and 45° Y-cut RS in one cal¬ 
culation, it is convenient to make the following 


changes in Figure 8: (1) reindex all elements 
in accordance with their positions (strictly the 
reindexed elements should carry double primes 
or some other designation, but for convenience 
these are dropped) ; (2) drop the factor o, 
merely remembering that the affected elements 
all change sign together. These changes lead to 
the matrix shown in Figure 9. 

Nothing has so far been said about the nu¬ 
merical values of the various elements involved. 
If we knew the numerical values referred to the 
crystal axes, it would be straightforward, but 
tedious, to evaluate those referred to the axes 
suitable to Figure 9. However, results obtained 
by Brush Development Company and Bell Tele¬ 
phone Laboratories are in significant disagree¬ 
ment, and we therefore prefer a different pro¬ 
cedure. 

It is our opinion, admittedly based on scant 
information, that the above discrepancies arise 
from attempting to evaluate the numerous 
quantities involved by exciting high shear 
modes. Both experimentally (silicon carbide 
dust pictures) and theoretically (Section 2.5.3) 


®ii ®’i3 ®i2 

<^*15 

<"'25 

®i3 ®'ll ®'l2 

«i5 

■'"»25 

c',2 c',2 0*22 

•^*25 

0 


«44 '"'46 

•'’14 •‘"f'34 


‘"'=46 '=44 

‘"'■34 '’|4 

acjj ac'5 ac'25 

'='55 



-'14 ‘"'34 

K', 


■'"''34 »‘,4 

+aK',3 K', 

‘^^25 ’*^25 0 


•^22 


Figure 8. Matrix of 45° Y-cut RS, referred to 
axes comparable to those shown in Figure 6. 


one must have serious doubts concerning the 
interpretation of such measurements. 

Calculations show (Section 2.5.3) that, as 
would be expected, only a small number of sub¬ 
determinants of the matrix in Figure 9 have a 
significant influence upon the motion of rec¬ 
tangular plates of 45° Y-cut RS and 45° Z-cut 
























RECIPROCITY; EQUIVALENT CIRCUIT 


59 


ADP. For the limited purposes of this volume, 
therefore, it seems best to carry out the calcu¬ 
lation without assuming numerical values but 
having used only the very general symmetry 
arguments to deduce the general structure, and 
then to evaluate the significant combinations of 
these elements by experiments intimately re¬ 
lated to the actual motion of a crystal in an un¬ 
derwater transducer. Thus, Figure 9 is solely 



Figure 9. Matrix of 45° Y-cut RS, including 
those of 45° Z-cut ADP as a special case. 


an indication of structure deduced from sym¬ 
metry. 


2^ RECIPROCITY; EQUIVALENT CIRCUIT 

The reciprocity principle is the basis of the 
absolute calibration of standard transducers 
and is, in addition, a very valuable design tool.“ 
It has been repeatedly verified experimentally, 
within a decibel or so, for crystal transducers 
constructed with 45° Y-cut RS and 45° Z-cut 
ADP. Since there is no evidence of any trend 
to the discrepancies, it is assumed that these 
arise from experimental error, and the prin¬ 
ciple is therefore regarded as valid. 

It therefore becomes important, as a check on 
the theory as so far developed, to determine if 
this principle is one of its consequences. The 
theory is summarized by the boundary-value 

® See Chapter 4 on some applications of the reci¬ 
procity principle to design. 


problem of Section 2.3.2, and it will now be 
shown that the principle is a direct consequence 
of this boundary-value problem. 

In a practical transducer, the external case 
is usually metal except for a rubber window, 
and hence very little of the electric field escapes; 
this is especially true in salt water or, in fact, 
any natural body of water except perhaps a 
very pure mountain lake. Consequently, the ex¬ 
ternal bounding surface in Figure 3 of Section 
2.3.2 may be any surface from that just enclos¬ 
ing the transducer to the surface at infinity. 
Also, there is nothing to prevent the region R 
from containing two or more transducers, since 
the number of conductors is not specified; in 
fact, any one of the elastically excluded im¬ 
pedance regions could be a transducer, since it 
was not assumed that these regions were ho¬ 
mogeneous or that they contained no conduc¬ 
tors. 

Now let the continuous distribution of force 
and impedance over the impedance surfaces, in¬ 
cluding the external surface, be replaced by a 
large number of pistons, numbered 1, 2, . . . :n: 
. . . , each very small compared to the shortest 
wavelength under consideration so that the 
active force (per unit area) and stiffance may 
be regarded as constant over each piston. This 
approximation may be made arbitrarily good, 
and it avoids the occurrence of integral equa¬ 
tions. 

Assuming that all material constants and the 
surface stiffances on each piston are known, the 
behavior of the system is completely determined 
by the potential amplitudes at which each 
conductor, and the force F (per unit area) 
at which each piston, is driven. The generalized 
displacements corresponding to these general¬ 
ized forces may clearly be taken as the charge 

on each conductor and the average displace¬ 
ment of each piston (multiplied by the area of 
the piston, for convenience), 

^Trqc"= J* dSupDp (Wp outward), (178) 

c" 

Up"." = jdSup". (179) 

Eliminating explicit reference to the details 
of the internal motion, these generalized dis- 













60 


BASIC THEORIES 


placements are linear functions of the general¬ 
ized forces 

qc” = C."e'Vo' + (180) 

1/p'V" = + Cp"."pV'F^v'. (181) 

The coefficients are generalized compliances and are 
functions of the material constants, the piston stiff- 
ances, the geometry of the system, and the frequency. 
Their interpretation is as follows. Cc"c' is the free- 
piston capacitance matrix; Cc"p'-k' is the short-circuit 
response to external forces; Cp'^-'c' is the free-piston 
transmitter response; and is the short-cir¬ 

cuit response to mechanical drive. 

Our problem is to show that it is a conse¬ 
quence of the boundary-value problem that the 
above compliance matrix is symmetric and 
hence that the above set of equations is circuit¬ 
like and subject to all the general circuit 
theorems. The impedance matrix is (1/fco) 
times the reciprocal of this compliance matrix; 
its existence is assured by the physical consider¬ 
ation that by forcing definite charge and dis¬ 
placement amplitudes upon the system, deter¬ 
minate potentials and forces must result and 
hence the above set of equations must be 
solvable for V^' and FpW 

The proof that this symmetry is a conse¬ 
quence of the boundary-value problem is closely 
analogous to that for a pure dielectric or a pure 
elastic system. One considers the special solu¬ 
tion Uj^{c) corresponding to unit potential 

amplitude on one conductor, zero on all others, 
and zero forces on all pistons. As c ranges over 
all conductors, this yields a set of free-piston 
solutions. Similarly, one considers the solution 
Upip'jt'), cfiip'n') corresponding to the p'-com- 
ponent of the force on the jr'-piston being unity, 
all other components on this and all other 
pistons, and all potentials, being zero. This 
yields a set of grounded-conductor solutions. 

By superposition of these special solutions, 
one obtains the general solution 

Up = Up{c')Vc' + Up{p'Tr')Fp\', (182) 

cl, = «/>(c')Vo' + (/>(pV)F,v'. (183) 

The auxiliary field quantities corresponding to 
each of the special solutions are called (c), 
Fj^^j(c), etc., and these quantities obey the same 
superposition rule. 


It is now readily shown that the elements of 


compliance matrix are given by 


Ce"e' = 

J dSupDpic'), 

(184) 

f ,, ,, , 

V' = J dSup"(p'F), 

(185) 

C.-'p. 

' = J dSnpDpip'F), 

(186) 


= J dSup'ic"). 

(187) 


It must be shown that the antisymmetric part 
of equations (184) and (185) are zero, the first 
with respect to exchange of the indices c" and 
c', and the second with respect to exchange of 
the pairs p'V' and pV; and that the electro¬ 
mechanical coupling compliances, equations 
(186) and (187), are equal. This is straight¬ 
forward but a little long, and will therefore be 
done in detail only for Cc"c'; this, together with 
an outline for the others, will illustrate the 
method so that the others can be readily treated. 

Since (^(c") is unity on c", and zero on all 
other conductors including the ground, one can 
insert it under the integral in equation (184) 
and then extend the integral over all conductors 
and the external surface. The resultant flux out 
of all conductors after antisymmetrizing, can 
then be transformed to a volume integral, 

47r(Co"e' - Co'e") 

dV[Dp{c')cl>(c") - Dp(c")<t,{c')lp. (188) 

U—c 

The divergence of is zero, so that two 
terms fall out in expanding the divergence 
bracket; also the result vanishes inside the con¬ 
ductors because there is no electric field there, 
and hence the region of integration may be ex¬ 
tended to include the interiors of the conduc¬ 
tors, which are a part of the elastic domain. Ex¬ 
pressing Zip in terms of the strain and field, the 
two bilinear forms in the field cancel and one 
is left with 

C.".' - C.'," 

= -/,„J dV[u,,.(c')4>.Ac")-u,Ac")4..,{c')], (189) 

R—Z 





AN EQUIVALENT VARIATIONAL PRINCIPLE 


61 


in which the removal of the interior impedance 
regions is permissible because f vanishes there. 

If the system were nonpiezoelectric, this 
would complete the proof, since the f tensor 
would then be zero. However, we now eliminate 
fprs^,p(^') ^,(c") with the stress equa¬ 

tion. A pair of bilinear forms in the strains 
cancel, and one is left with 

Ce"o' - Ce'e” 

= -j'dV[F„(c")uUc')-Fr.(c')uUc")]. (190) 

R — Z 

The first term inside the bracket is 

{Fr.{c")Ur{c') ]., - Frs..{c")Ur{c'). (191) 

Eliminating the divergence of the stress tensor 
with the propagation equation, the second term 
of equation (191) becomes 

—pco%(c")M,.(c'). 

The resultant dot product of the two displace¬ 
ments is symmetric and hence falls out upon 
exchange of the indices, and we are left with 
an exact divergence which, upon being con¬ 
verted to a surface integral over the impedance 
surfaces and the external boundary, that is, 
over all the pistons, yields 

Cc"o' - Ce'o" 

dSn^lP^,{c")u,{c') -Pp,(c')w,(c")]. (192) 

Now and c') are the stresses, corre¬ 

sponding to free-piston solutions, and hence 
their flux across the pistons are from the bound¬ 
ary conditions, u^{c" or the above 

bracket therefore becomes the difference of two 
bilinear forms in u^{c') and u^ic") and, since 
Spg is symmetric, vanishes. 

In treating equation (185), one replaces 
Up" (pV) by 

up(p'^') [Pp,(p"^")n, + Sp,u,(p"^") ], (193) 

and extends the integral over all pistons; this 
is permissible since the bracket is zero on every 
piston except n" and is zero there unless is 
p", in which case it is one. Then the stiffance 
term falls out upon exchange and the remaining 
exact flux can be converted to a volume integral 
over R — Z. The divergence is expanded, the 
displacement propagation equation used, and 


the stress expressed in terms of the strain and 
field. This latter brings out an / which, vanish¬ 
ing inside the impedance regions, allows the 
region to be extended from R — Z to R. Having 
served its purpose, the / tensor is eliminated 
with the equation for the electric displacement; 
the result is an exact divergence which can be 
converted to a surface integral over all con¬ 
ductors, and here the special </)(p";i") and 
0(p'jt') vanish. 

In treating equations (186) and (187) one 
finds it most convenient to bring equation (186) 
to an integral over the pistons, by methods simi¬ 
lar to those described above, whereupon it be¬ 
comes identical to equation (187). 

The foregoing shows that a linear dissipative 
piezoelectric system, governed by the boundary- 
value problem of Section 2.3.2, has the equiva¬ 
lent circuit shown in Figure 10. 



Figure 10. The rigorous equivalent circuit for 
a linear dissipative piezoelectric system. 


2 5 an equivalent variational 

PRINCIPLE 


2.5.1 Steady-State Boundary-Value 

Problem 

The steady-state solution of the above bound¬ 
ary-value problem is adequate for our purposes, 
since the transient behavior can be determined 
by superposition of steady states. This problem 
is defined by 

pw-Up -f = 0 (in the crystal), (194) 
div D = 0 (in all space), (195) 

P pqTlq -j- S pqllq = Fp H 

(over surface of crystal), ^ 














62 


BASIC THEORIES 


surf div D = 4x0- nQ7'> 

(on every surface of discontinuity), ^ 

4) = (constant on electrodes, n 
continuous everywhere), 

m which the displacement, strain, etc., are com¬ 
plex amplitudes. The complex tensor acts on 
the displacement instead of the velocity, and 
may be conveniently called the “stiffance”; it is 
just in which is the ordinary (veloc¬ 

ity) impedance tensor. Its value for various 
surfaces and frequencies is a set of parameters 
to be experimentally determined a posteriori; 
for the present, we need only assume that it is 
symmetric and has only two distinct principal 
values, the normal and tangential specific acous¬ 
tic (displacement) impedances, as discussed 
previously. 

The solution of this problem by the character¬ 
istic function method is hopelessly complicated, 
since, to mention only one complication, the 
characteristic values will be solutions of some 
set of simultaneous complex transcendental 
equations. 


Variational Principle 

If, however, we can find a variation principle 
equivalent to this problem, we can use the pow¬ 
erful direct method, of which the well-known 
Rayleigh-Ritz method of treating conservative 
systems is a special case. This method has many 
advantages, the most important being that we 
can use our qualitative understanding of a 
physical system to guide us in the choice of trial 
solutions. 

The derivation of the equations of motion of 
a dissipative system from a variation principle 
was first accomplished many years ago by 
Bateman,but the great power of this 
method for treating complicated practical prob¬ 
lems does not seem to have been appreciated. 
Some extensions of Bateman’s valuable contri¬ 
bution, with applications to various acoustic 
problems, will shortly be published elsewhere. 

Leaving aside the arguments which lead us 
to choose the following variation principle, con¬ 
sider the integral 


I = 


+ 


J dS[FpUp - ^UpSp^Ug - (0 - V)cr], 


W' = is'cs - + sfE. 

OTT 


( 200 ) 


The variational principle 6/ = 0 will be shown 
to be equivalent to the boundary-value problem 
governing the behavior of a transmitter, and it 
will lead to an equivalent circuit for this case. 
The equivalent circuit for a receiver will then 
be obtained by reciprocity considerations (see 
Chapter 3). 

The volume integral extends over all space, 
but only the terms in the electric field will make 
contributions outside the crystal because c and f 
and the displacements are zero there; the sur¬ 
face integral extends over all surfaces of dis¬ 
continuity, but here again only the electric-field 
terms will make contributions outside the 
crystal. 

The total integral / is a complex number 
whose value depends upon our choice of </>, u^, 
and o since we assume that all constants, the 
external surface force density (zero for a 
transmitter), and the potentials V on all metal 
surfaces, are given. It will now be shown that 
amongst all functions <f>, u^, and a, that set which 
gives I a stationary value satisfies equations 
(194) to (198), so that demanding that the 
first variation of / (under arbitrary variations 
of </>, and o) shall vanish is equivalent to the 
complete boundary-value problem. This station¬ 
ary value is of course not an extremal, since 
there is no meaning attached to the extremal of 
a complex function; but neither does Hamilton’s 
principle correspond to an extremal, and yet its 
usefulness is well known. 

The first variation of I is 

dl = J* dV(po}-Up8up — 8W') 

^ dS [ Fp 8 Up — 8 ll pS pqllq — cr50 — (</) — V) 8 (T ]. 

( 201 ) 

The function W' is simply related to, but not 
identical with, the energy density W. It is easily 
shown that its variation is given by 


8W' = Ppq8Up,q + 


Dr8<i>,r 



( 202 ) 







AN EQUIVALENT VARIATIONAL PRINCIPLE 


63 


Inserting these values into equation (201), 
and doing two partial integrations according to 


Then, under arbitrary variations of Up, </>, and o 
as before, but not varying Do, SJ is given by 


P pqblL p q i^P pq^U p) _q P pq.q^^ pj 

and 

8(t)rDr = div (D50) — 84) div D, 

we find 

Si = J' dV^ 8Up{pocrUp + Ppq,q) + 50 div ^ 

+ J" dS 1^ 8Up{Fp - SpqUq - PpqRq) 

+ 50(surf div - a) - (0 - y)5(r J. (203) 

We see that the principle 6 / = 0 for arbitrary 
variations of 0 , Up and o is exactly equivalent to 
equations (194) to (197), and we may there¬ 
fore apply the direct method to I, equation 
(199). It is to be emphasized that the equa¬ 
tions of state are assumed, so that Pp^ and 
Dr are expressible in terms of the derivatives 
of the field quantities, but that the propagation 
equations and all boundary conditions, except, 
of course, equation (198), are consequences of 
the variational principle hi = 0. Thus, a, whose 
integral over the electrodes is the “displace¬ 
ment” conjugate to the externally driven poten¬ 
tials V, is varied independently and the vari¬ 
ational principle makes it match surf div D/4:n: 
on the electrodes. 

In applying the direct method to 1, we should 
assume a trial value for 0 out to infinity, or 
more correctly, out to the grounded closed con¬ 
ductor consisting of the case with rubber win¬ 
dows electrically closed by sea water. Such a 
treatment would not only take account of flux 
fringing near the edges of the crystal plate, but 
also of stray capacitances and dielectric losses 
shunting the crystal electrodes. However, it is 
not clear what to assume for the external poten¬ 
tial, and hence we resort to another expedient. 

Suppose the entire problem were solved, so 
that we knew the external electric displacement, 
Do. Let 


J = JdV(po)^-^ 

crystal 

volume 


-W) +J'dS(F, 

all crystal 
surfaces 




(204) 


+ J dSa(.V -■#■)+/ 

plated unplated 

surfaces surfaces 


in which n is the outward normal to the crystal. 


8<J — J' dV 8Up{pU~Up + P pqq) + 60 ^ ^ 

crystal 
volume 

- j* dS 8Up{SpqUq + Ppqllq - Fp) 

all crystal 
surfaces 

+ Jrfs[a,T(V- ^,) + ~ U ^ 

plated 
surfaces 



+ / 

unplated 

surfaces 


(205) 


The principle 6 / = 0 now yields the proper 
propagation equations and boundary conditions 
but contains the unknown vector Do in the last 
term of equation (205). 

It can be shown that the error arising from 
dropping the entire last term is negligible in 
the case of 45° Y-cut RS and 45° Z-cut ADP, 
if the electrodes fully cover the electrode faces. 
It should be noticed that hJ' the result of drop¬ 
ping the last term in equation (205), is not the 
exact variation of any integral, since it can be 
obtained by dropping Dq in J, equation (204), 
taking the variation, and adding to this a term 


J ( 206 ) 

unplated 

surfaces 

to cancel what is left of the last term of equa¬ 
tion (205). This addendum is not an exact vari¬ 
ation since D depends upon the derivative of 
the displacement and potential. 

We are therefore left with only one error of 
consequence, which arises from the fact that 
the a deduced from equation (205) is the sur¬ 
face density only on the face of the electrode 
in contact with the crystal. Thus we will make 
an error in computing the current amplitude 
onto the electrodes. However, this error will 
correspond almost exactly to the effect produced 
by a pair of condensers, one from each electrode 
to the case. The error in this correspondence is 
of the same order as the error in neglecting the 
last term of equation (205). 

Therefore, when we finally deduce an equiva¬ 
lent circuit, we need only insert these condens¬ 
ers to get a very accurate equivalent circuit for 









64 


BASIC THEORIES 


the single crystal. The capacitances of these 
condensers is not given by equation (205) but 
they can be measured and, unless restricted 
space requires the electrodes to come quite close 
to metal parts of the case, are not of great prac¬ 
tical importance. In any event, their effect can 
be taken into account. 


Solution of the Boundary-Value 
Problem 

The boundary-value problem governing the 
motion of a loaded piezoelectrical crystal was 
formulated in Sections 2.3.1 and 2.3.2. In Sec¬ 



tion 2.5.2, a variational principle rigorously 
equivalent to this boundary-value problem was 
presented. 

This variational principle will now be applied 
to a single rectangular plate of 45° Y-cut RS 
or 45° Z-cut ADP. The solution is first obtained 
in the Mason approximation, and in the next 
section, in a higher approximation. 

Consider a rectangular plate of 45° Y-cut RS 
or 45° Z-cut ADP, referred to the xyz frame 
to which the matrices of Section 2.3.3, Figure 9, 
refer. This coordinate frame, shown in Figure 


11, permits 45° Y-cut RS and 45° Z-cut ADP 
to be treated in one calculation, when the ap¬ 
propriate matrices are used. 

The displacement and the potential is as¬ 
sumed to be expansible in a power series in the 
lateral coordinates x and .e, with coefficients that 
are arbitrary functions of y, 

M = Wo + UiaX + UqiZ -f W20X- + UnXZ + Uq-iZ- + • * ', 
V = Vo i^io^ + • • •> ( 207 ) 

W = Wo WioX + • • •, 

= <^0 + 010 ^ + • • •. 

The usefulness of the Mason approximation, 
which will be shown to be equivalent to such an 
expansion in which only Uio, Vo, and Uoi are dif¬ 
ferent from zero, is sufficient evidence that 
there is a significant domain in which the above 
expansion is valid. Physically, one would expect 
the expansion to converge rapidly if the thick¬ 
ness and width are small fractions of the longi¬ 
tudinal wavelength as deduced from the Mason 
approximation, and the ratio of thickness to 
width is sufficiently small so that the parallel- 
plate condenser approximation is good. 

On the other hand, there is ample experi¬ 
mental evidence to show that errors of serious 
practical significance to the design of trans¬ 
ducers occur if one does not save a few of these 
terms. In order to gain some idea of the com¬ 
plexity of the motion of crystal plates, it will 
be helpful to study Figures 12 and 13, which 
are photographs of the distributions of silicon 
carbide dust which develop when the crystals 
are driven at various frequencies, in air. 

In Figure 12 , all the crystals are driven at or 
near their lowest (free-free) resonance. A study 
of these photos brings out three important 
points: ( 1 ) even near resonance, the motion is 
more complicated than that envisaged in the 
Mason approximation; ( 2 ) the motion of RS 
is considerably more complicated than that of 
ADP, as suggested by the more complicated 
matrices of RS compared to ADP (see Figures 
6 and 7 of Section 2.3.3) ; and (3) the motion 
of both RS and ADP becomes more complicated 
as the width-length ratio increases. 

In Figure 13, the crystals are driven at or 
near their second, third, and fourth resonances. 
The photos give only a partial idea of the 
enormous complexity of the motion. For exam- 















65 


AN EQUIVALENT VARIATIONAL PRINCIPLE ' 





Figure 12. Silicon carbide dust patterns of 45° Z-cut ADP and 45° Y-cut RS at their fundamental 
resonances. 















66 


BASIC THEORIES 





45* Z-CUT ADP 
XI IN. 


A SECOND RESONANCE 


B THIRD RESONANCE 


C FOURTH RESONANCE 


45* Y-CUT RS 
IN. 


SECOND RESONANCE 


THIRD RESONANCE 


FOURTH RESONANCE 


Figure 13. Silicon carbide dust patterns on electrode faces at higher resonances. 






AN EQUIVALENT VARIATIONAL PRINCIPLE 


67 


pie, the circular distributions in Figures 13B 
and 13C are actually vortices of silicon carbide 
dust, circulating continuously. To treat crystal 
plates near these higher resonances, one will 
presumably need to keep several terms of equa¬ 
tion (207). 

In order to understand the physics of the 
calculation which follows, it is helpful to study 
the modes of motion that would result if all but 
one of the displacement terms, in the expan¬ 
sions of equation (207), were zero. An indica¬ 
tion of these motions is given in Figures 14A, 
14B and 14C, and one sees that a free crystal 
could not vibrate in any one of these modes; 
that is, they are not normal modes. In fact, it 
would require an intricate system of blocks and 
external stresses to make any one term a normal 
mode. Since no single term is a normal mode, 
we must expect to find that, to some approxi¬ 
mation not yet known, certain sets of these 
terms will be rigidly coupled together by sets 
of quantities which are generalizations of Pois¬ 
son’s ratio. This notion is very helpful in sug¬ 
gesting an attack on the equations obtained by 
applying the variational principle of Section 
2.5.2 to the expansions of equation (207). 

Our general procedure is as follows. Entering 
the variational principle with equation (207), 
one can perform all x and ^ integrations ex¬ 
plicitly and thus obtain a variational principle 
in which the volume and surface Lagrangians 
are expansions in the moments of all orders" 
of an a;, 2 :-section, with coefficients which are 
functions of y only. Thus, the four field quan¬ 
tities u,v,w, and cf>, functions of three variables, 
have been replaced by an infinite set of field 
quantities tio, ^^lo, etc., which are, however, 
functions of only one variable, y. One advantage 
of the variational principle is now seen more 
clearly: we could deduce the coupled equations 
satisfied by the Uo, Uio, etc., from the funda¬ 
mental field equations, but we would not know 
how to weight the successive equations. The 
variational equation weights these field equa¬ 
tions, in a unified way, with successively higher- 
order moments of the cross section and we are 
thus able to make a consistent approximation. 
It is advantageous to refer to this procedure of 

n Because of the rectangular shape, all odd moments 
fall out in our present problem. 


replacing the dependence upon some of the in¬ 
dependent variables by an expansion in an inde¬ 
pendent set of functions, while keeping the 
dependence upon other variables in field quan¬ 
tities, as the semidirect method. This method 
stands midway between the original variational 
principle, which merely generates the field 
equations (and, in the present case, the bound¬ 
ary conditions), and the complete direct method 
invented by Ritz. 


Mason Approximation, with a Modification 

We will gain considerable facility with the 
variational treatment, and also test the method, 
by first showing that it gives the right answer 
in the Mason approximation. Accordingly, we 
assume that the principle longitudinal mode 
(vo) is coupled only to Poisson motions in the 
two lateral directions, and that the potential is 
that of an infinitely thin parallel-plate con¬ 
denser. 


U = UioX = Ux 

V = Vo = V 

W = WqiZ = Wz 

(f) = (polZ = 4>2 


(208) 


Inserting these trial functions into the vari¬ 
ational principle together with the assumption 
that a, the surface charge density, depends upon 
y only and merely reverses sign in going from 
one electrode to the other, one finds 


A 



dU{poi^X^U-CnU-CuV -CizW -t-/3i4>) 


- 8U'CooX^~U + 8Vpo:W - 8V'P 


-f 8W{po:^zW - CnU - CnV' - CzzW) 

^ - fn(U - V') - t ] 


— 8W'cuZ^W -f 64> 


+ - (iVNVh - (SVNVh 

+ (209) 

P = CnV + CnU + C„W + (210) 


All impedances have been neglected except the 
normal reaction at the two ends of the crystal, 
here represented by the normal stiffances Ni 
and N 2 . 

Without doing the possible partial Integra- 






68 


BASIC THEORIES 


tions on the W' and 6 IE' terms, we further spe¬ 
cialize U and IE so that the coefficients of bU 
and 8 IE vanish, obtaining 

CnU + C 13 IE = / 3 i<I> — C 12 V' + porX~U, /o-i 1 \ 
Ci3t/ + C 33 IE = 0 - 013^' + PCOW. 

This is solved in 0th order (i.e., neglecting x- 
and z-}, the result placed in the right member, 


To dispose of the bU' and 8 IE' terms, we use 
the solution of equation ( 212 ) only to 0 th order, 
so as to consistently keep terms only up to the 
second-order moments. The bU' term then be¬ 
comes 

- 8U'C,,'^-U' = - 5y"c666|^V", (215) 

with a similar result for the 8 TE' term. This 


A 



Y 










































Zi 







Y 




Gnm 









Figure 14. Motions corresponding to individual terms of equation (207). 


mm 



and thus a second approximation is obtained, 
which neglects terms in the fourth-order mo¬ 
ments. Only the linear combination P, which 
represents the effective stress in the y direction, 
is involved in this problem if we save only 
second-order moments, and this is 

p = YV' + Fcf, (212) 

y = Cn - + ejz-), (213) 

jP = [1 + pur{txAi\X- + €jA.i 32 “)]/ 3 i, (214) 

in which An and A 13 are elements of the recip¬ 
rocal of the matrix of equation ( 211 ), and 
and E, are the Poisson ratios in the x and ,t: 
directions. 


term combines with the 8 y term because, in 0 th 
approximation 

y" = - 

12 _ (216) 

Ao - 

The surface 8 <^I> terms combine with the vol¬ 
ume term, since <!> has been assumed independ¬ 
ent of y. However, these surface terms are 
completely negligible in all practical cases; it 
is fortunate that they are negligible because, 
owing to the assumed form of the trial displace¬ 
ments and potential, they destroy reciprocity, 
so that if they had any appreciable effect, a 





































































































































AN EQUIVALENT VARIATIONAL PRINCIPLE 


69 


more complicated set of trial functions would 
be needed.” 

Doing the partial integration on the 6U' term, 
one obtains for the field equations and boundary 


conditions 

+ P' = 0, (217) 

T = (218) 

p+ = p ; 1 - kUe'ix-C66 + e‘^Z-C44) [■, (219) 

= Kss + (220) 

P 2 + N 2 V 2 = 0, (221) 

-Pi+NiVi = 0. (222) 


First, if we neglect x- and z-, these are ex¬ 
actly Mason’s^-*^ equations, in a different nota¬ 
tion. Second, we see that, keeping the second- 
order moments, they are identical in form with 
Mason’s equations but their physical implica¬ 
tion is quite different because the various coeffi¬ 
cients are now functions of frequency. 

The identity in form allows us to use the 
Mason equivalent circuit for calculations, merely 
using the frequency dependent parameters in¬ 
stead of his constant parameters. It will be 
advantageous now to come to a physical under¬ 
standing of equations (217) to (222), and equa¬ 
tion (213). For this purpose, it is advantageous 
first to put equation (217) in a different form, 
one which is not as convenient for calculations, 
but which allows us to see what it means: 

- p++a,2V - YoV" = (e^^Cee + ei?Cu)V", (223) 
P++ = p j 1 -b koiejx^ -f- ejz-) ). (224) 

In equation (223), the first term is the kinetic 
reaction arising from all inertial effects, and 
from equation (224) we see that the effective 
density is greater than the actual by an amount 
which expresses the relative inertial effect^ cor¬ 
responding to the two Poisson motions which 
are necessary to relax the lateral stresses. This 
effect is of considerable practical importance in 

o These terms arise from the fringing of the electric 
displacement corresponding to a small shear strain in 
the assumed displacements. Reciprocity is restored if 
higher terms of V are included. See Section 2.4. 

p This effect, in isotropic systems, was discovered by 
Pockhammer.-*^- 


crystal transducers. The second term of equa¬ 
tion (223) is the ordinary internal body force 
arising from internal variations of the longi¬ 
tudinal stress. 

The right member of equation (223) is very 
interesting. The variational principle gives us 
the solution of the problem corresponding to 
the given driving forces together with any 
other extraneous forces necessary to maintain 
any constraints implied by the form assumed 
for the trial displacements. The trial functions 
include a shear strain supported by no surface 
forces; therefore the solution contains a ficti¬ 
tious body force to support this shear strain. 

We therefore conclude that equation (223) 
is incorrect, in second order, for the problem 
of interest, because it contains only a part of 
the terms containing second-order moments. 
Thus, the foregoing treatment, although giving 
the very important lateral inertia term cor¬ 
rectly, is valid only in the Mason approximation 
in which all second-order moments are neg¬ 
lected. 

Approximation Including All 
Second-Order Moments 

To include all terms containing second-order 
moments, we take more terms in the trial func¬ 
tions. The details of this calculation are quite 
lengthy, and hence only the essential points will 
be given here. 

First, we assume that tangential and normal 
stiffances act over all faces. The problem is 
greatly simplified, without losing the essential 
features, if we assume that the stiffances have 
the same value at all points on the lateral faces, 
so that the lateral stiifance is specified by the 
normal and tangential characteristic values N 
and T. 

On the ends we assume normal stiffances Ni 
and No which, except for the factor io) (neces¬ 
sary to convert an impedance to a stiffance), 
are identical to the radiation impedances as¬ 
sumed by Mason. For the present, we neglect 
the tangential stiffances on the ends because 
these bring in a new kind of term and destroy 
the simple result to be proven, that without 
these terms we can use the Mason circuit by 
merely allowing certain parameters to vary 
with frequency, width, etc. The error so intro- 




70 


BASIC THEORIES 


duced is a term which vanishes with the second 
moment and this is likely to be of practical im¬ 
portance in only one case, that in which one 
face is cemented to a backing plate by a cement 
so rigid as to seriously oppose tangential mo¬ 
tion. The theoretical study of this case, perhaps 
of considerable importance, is not contained in 
this volume. 

We keep all effects not considered in the 
Mason approximation only to the lowest order 
in which each occurs. Subject to this, and the 
neglect of tangential stiffances on the end faces 
as noted above, we keep all terms in the vari¬ 
ational integral involving second-order mo¬ 
ments, but drop all higher moments (because 
of the symmetry of a rectangle, the third-order 
moments vanish so that we are dropping fourth 
and higher moments). 

To be sure of getting all second-order mo¬ 
ments, we must keep cubic terms in the expan¬ 
sions of the displacement and potential, since 
these latter are differentiated once to get the 
strains and electric displacement. Actually, on 
account of the symmetry structure of the mat¬ 
rices of RS and ADP, the 45° cuts, and the 
symmetry of a rectangular plate, the cubic 
terms enter only to a very slight extent. 

Instead of a simple power series expansion, 
it is found convenient to expand in a set of 
orthogo'nal functions XaZ^ of x and z, defined 
by 

X = ^ r ~ - 3x7^) . 

“ 2 ’ 6 ’ (225) 

(a = 0 to 3), 

and similarly for Z^. In addition to being or¬ 
thogonal, vanishing like the ath and (3th power 
of the width and thickness, respectively, these 
functions have the additional property, very 
convenient for calculation, that 

= Xa-u with X_i = 0, (226) 

and similarly for dZ^/clz. As a convenience to 
notation, we use the same symbols Mo, etc., 
as in the previous section, although they now 
have slightly different meanings. 

The foregoing refers only to the expansion 
of the displacement. It can be shown that an 
adequate form for the potential, and one more 


convenient for calculation since it simplifies so 
nicely on the electrodes, is 

(f) = + {z- — zf) {\po + \piQX + ^Qiz). (227) 

Finally, the surface charge density is ex¬ 
panded according to 

(228) 

a=0 

on the ±Zi plates, in which the various coeffi¬ 
cients are functions of y only. As seen at a 
glance from the only term in the surface La- 
grangian which involves o, all ot above a = 0 
fall out completely. 

Finally, although the variational principle 
will yield the result, we can save some trivial 
complications by assuming at the outset that (/>! 
is actually independent of y and that = ±oo. 

These trial functions are now inserted into 
the modified principle hJ' = 0. The labor of 
forming this expression is greatly reduced 
by using the Einstein summation convention, 
whereby Ua^XaZ^ represents the sum of all 
terms as a and (3 range from 0 to 3; the deriva¬ 
tive recursion formulas, equation (226) and 
the ortho-normality conditions satisfied by Aa 
and Z^, 

Examining the variational equations one finds 
that they are linear algebraic to the present 
approximation, except for the one governing Vq. 
Also, the algebraic set falls apart into subsets, 
which greatly simplifies their solution. Thus 
certain of the displacement coefficients are lin¬ 
ear combinations of a particular one and the 
electric field ^i, while others satisfy linear ho¬ 
mogeneous equations with nonvanishing deter¬ 
minant. Physically, this means that certain 
motions, in the present approximation, are 
rigidly coupled to others by a generalization 
of a pantograph mechanism, the coupling con¬ 
stants being an array of quantities which are a 
generalization of Poisson’s ratio; others, satis¬ 
fying the homogeneous equations, form a sep¬ 
arate physical system free to vibrate independ¬ 
ently, but are not excited in this order. 

An example of a homogeneous set is 

044(1^01 + u ) q ) + 045(^01 + u^io) = 0 , ( 229 ) 

Cibivoi + n^o) + C55(woi + iUio) = 0. 

Since no subdeterminant of the c matrix can 






AN EQUIVALENT VARIATIONAL PRINCIPLE 


71 


vanish (the strain energy is positive-definite), 
one concludes that the parentheses are zero.^i 

Another set takes the form 

C 11 W 20 + CnWn Ci 6 (Wio + t^ 2 o) = ~ CuVio, 

C 13 W 20 + CssWn + C36(wio + V 20 ) = — Ci3l'l0j (230) 

C\6U2o + Ci^Wn + C 66 (Wio + 1 ^ 20 ) = — Cierio- 

Using the abbreviations C for the 3x3 matrix, 

and U2 and c for the 3x1 matrices wdth elements 
^ 0 , etc., and Cn, etc., equation (230) becomes 
C112 = —cviQ. The solution of this is Uo = 
—Efio', in which e is a 3x1 satisfying 

Ce = c. (231) 

The three elements of e will be called e^., e^ and 
E^, the subscripts meaning width, thickness, and 
shear. If the elements Ck? and Cse of C, which 
couple compressional and shear strains, are 
zero (as they are for ADP but not RS), then 
E,^, and E, are the ordinary Poisson ratios used by 
Mason. The e’s are therefore generalizations of 
the Poisson ratios. 

Another set, also involving e and one which 
allows the physical meaning of this matrix to 
be more clearly seen, is 

CiiMio+Ci 3Z/;01+016^10 = -Ci2i;o-/31</>1 + 0^), 

C 13 W 10 +C33i^oi +C36i^io = -Ci3i;o + 0 + 0(^), (232) 

CieMio +C36Zt’oi d-Ceefio = ~Ci^Vo + 0 + 0{x^), 

which yields 

Ui = — e'Uo + (C^i, O(V^), 

in which Cl} means (C“^)ii, etc., the designated 
element of the reciprocal matrix. It is interesting 
to notice that the same matrices C, c and e 
connect U2 to v'lo and Ui to v^. 

Now if Cio and c^r, are zero, equation (232) is 
the same as that which couples the Poisson 
“breathing,” and Woi, to the principal longi¬ 
tudinal motion Vo, together with the equation 
Via = 0. However, it is seen that Vio is also 
coupled to Vo in RS. Thus the electrode faces 
vibrate between two limiting parallelograms in 
the case of RS, but not in ADP. 

Taking account of all the algebraic equations, 
one finds that Uo, tvo, (tioi + ^^m), (^’o 2 + 

?<'„/) and (Uo> -f ?(’ii) are all zero. The values 
of other secondary displacement coefficients 
may become important when sufficiently de¬ 
tailed experimental data are available, and are 

■^1 Throughout this section, “zero” means “contributes 
terms to bJ' of order or higher.” 


readily calculated; however, for the present 
purpose, it is sufficient to use the relations to 
reduce the system down to a perturbed Vo prob¬ 
lem. Two other items of this reduction should 
be mentioned: First, in many of the terms of 
type dUa/ Q, the quantity Q vanishes by virtue 
of the algebraic relations, but some terms of 
this type remain; in all cases except the bv'o 
term, these are not partially integrated, but 
transformed to underived variations with the 
aid of the generalized Poisson relations and the 
solution of the Oth-order problem. Second, the 
auxiliary potential quantities olio, and if'oi are 
all zero in this approximation. 

Having disposed of all the algebraic relations, 
one is left with a propagation equation and 
boundary conditions for Vq (hereafter called 
V ), and an “equation of state” giving the charge 
density in terms of V' and A, the potential dif¬ 
ference between the plates. These are 

p+co^F -f y+U” = 0, (233) 

Ft 4- N 2 V 2 = 0, (234) 

-F|+A^iUi= 0, (235) 

(TO = ^ - F+V'. (236) 

The auxiliary quantities here introduced are 


defined by 




p+ = p 

-j 1 — — 

(^y)(“w )[’ 

(237) 

y+ 

= y 1 1 - k^X^Ll + ejj + 2 - 6 ? 



+(f)[4" 

+ e|j + fejJ j-. 

(238) 


p+ = YV' 

F+A 
t ’ 

(239) 

+ 

II 

;i 1 1 + Cw + X' 



|pco2 - 

-^) + A,C5 


(240) 

+ 

II 



. (241) 


k C 

c 

-(?)■ 

(242) 


in which iv and t are the width and thickness 
of the plate, and q+, etc., is the perturbed pa- 





72 


BASIC THEORIES 


rameter corresponding to q, etc. The quantity 
designated as O(x-) in equation (241) is of no 
importance in ADP or RS, and is therefore not 
written out. 

These equations define a boundary-value 
problem identical in structure with Mason’s, 
the different physical results arising solely from 
the dependence of the quantities p+, Y +, F+, and 
K+ on frequency, width, and thickness, and 
the stiffances T and N. This is a very convenient 
circumstance, since it allows the valuable con¬ 
cepts and results obtained by Mason to be taken 
over, merely applying a correction term here 
and there. 

Concerning the confidence which should be 
placed in the validity of this treatment, the 
following remarks are pertinent: 

1. The final results are consequences of well- 
known and accepted principles of physics to¬ 
gether with a variational principle which has 
been shown to be an alternate expression of 
these principles. 

2. The Mason theory, which has been the 

foundation of all design work, is contained in 
the final result as the special_case in_which the 
dimensionless quantities k~z\ T^YkH, 


and Nw/Y are negligibly small compared to 1. 

3. By setting T/Yk-t and Nw/Y to zero, but 
keeping the conservative x- and terms, the 
finite-width correction to the resonant fre¬ 
quency is correctly given, as will be shown in 
Chapter 3. 

It is therefore believed that the foregoing re¬ 
sults lay the basis for a much more detailed 
understanding of crystal transducers than is 
at present available. Before full use can be made 
of this new theory, a carefully planned program 
of experimental work to evaluate the param¬ 
eters T and N, especially their imaginary (dis¬ 
sipative) parts, will be necessary; the pressure 
of work during World War II did not permit 
this. 

However, some of the consequences of this 
theory can be discussed on the basis of available 
experimental data, and certain further conclu¬ 
sions can be tentatively reached concerning the 
qualitative character of dissipation. In Section 
2.4, it was shown that the above boundary-value 
problem is representable by an equivalent cir¬ 
cuit, and in Chapter 3 those aspects of the 
theory which are at present capable of experi¬ 
mental examination are discussed. 






Chapter 3 

PROPERTIES OF THE COMPONENT PARTS OF CRYSTAL TRANSDUCERS 

By Richard Bellman, T. Finley Burke, Glen D. Camp, Bourne G. Eaton, and Fred M. Uber 


I N THIS CHAPTER, ail attempt is made to come 
to as thorough an understanding as possible 
of the properties of the component parts of 
crystal transducers, and to study the couplings 
introduced between these parts when they are 
assembled into a completed unit. 

The first section begins with what one sees, 
a “black box” with a few feet of cable, and 
systematically dissects it, laying the parts aside 
for study in later sections. Wherever feasible, 
the qualitative properties of these parts will be 
briefly mentioned; also, an attempt is made to 
catalog the couplings between parts as they are 
separated. At the close of this section, one will 
have not only a collection of component parts 
requiring further study, but also a qualitative 
picture of the primary motivating element, the 
crystal, obscured by a number of electric and 
mechanical shunts. Each of these shunts is an 
obstacle to delivering power to, or collecting a 
signal from, the crystals. 

The remaining sections of this chapter are 
devoted to a detailed study of the parts them¬ 
selves. 


3 » DISSECTION OF A TRANSDUCER 


Cables 

By removing the packing gland, the cable is 
disconnected from the transducer terminals. 
The cable is a three-conductor system, two leads 
surrounded by a shield. In all except extreme 
applications, the cable will be a very small 
fraction of an (electromagnetic) wavelength, 
and we can treat it as a simple six-terminal net- 



Figure 1. Approximate circuit for a shielded 
cable. 


work. Neglecting small resistive drops in the 
leads themselves, we can reduce this network 
to the three-terminal network shown in Figure 
1. Numerical data on the capacitances and dis¬ 
sipation factors for each of the elements are 
given later (see Chapter 5). 


Electronic System 

Starting with a complete sonar system, we 
first detach the electronic system, leaving only 
the upper end of a cable of two or more con¬ 
ductors above water. The electronic system may 
be a simple driver (transmitter), a simple am¬ 
plifier (receiver), or a much more complicated 
system for various special applications. Al¬ 
though not properly a part of the crystal trans¬ 
ducer, it is extremely important that the elec¬ 
tronic system and the transducer be properly 
matched. The design of electronic systems for 
use with crystal transducers is discussed later 
(see Chapter 5). 


Matching Network 

A crystal transducer having any appreciable 
dissipation or radiation resistance is capacita- 
tive at all frequencies, and therefore a coil can 
always be found which will make the impedance 
purely resistive at least at one frequency; be¬ 
cause of the frequency dependence of the react¬ 
ance, it is sometimes possible to tune a trans¬ 
ducer at as many as three nearby frequencies. 
The coil is the simplest type of matching net¬ 
works and in many respects is the best (see 
Chapter 5). In any event, the most complicated 
matching network is a four-terminal system 



73 








74 


COMPONENT PARTS 


composed of inert elements, and its function is 
to perform an impedance transformation be¬ 
tween the cable and what we shall refer to as 
the stripped transducer. The combination of 
cable and matching network is shown in Figure 
2. The question mark indicates the necessity of 



Figure 2. Cable and matching network. 


making some definite disposition of the shield, 
the most usual disposition being to ground the 
upper end and leave the lower end disconnected. 


Preamplifier 

A small receiver has low capacitance and 
therefore high impedance, and the capacitative 
shunts between the leads and to the shield of 
the cable may appear essentially as a dead-short 
across the output terminals of the receiver. To 
circumvent this situation, a preamplifier is 
sometimes built into the case of the transducer. 
We shall regard everything between the cable 
leads and the leads to the crystal motor as a 
preamplifier, even though it may contain an 
inert matching network also. Then the most 
general preamplifier is an active four-terminal 
network whose function is both to perform an 
impedance transformation and to raise the 
level. The design of such systems is discussed 
later (see Chapter 5). 


Cases 

Depending on the detailed construction of the 
transducer, we now either remove a rubber 
sleeve or take off a rubber window molded into 
a metal ring, etc., remove® the crystal motor, 
drain out any coupling fluid (e.g., castor oil), 

a Window-coupled units are an exception since the 
crystals are cemented to the inner face of the window. 


and define everything left as the case, including 
the window or sleeve. The case serves a number 
of functions: it prevents the crystals and elec¬ 
tric leads from getting wet, it protects the 
crystals from mechanical shock, and it serves 
as a means of support to the motor and other 
parts. It consists of a hollow shell, various sup¬ 
porting brackets, and cavities for containing 
matching networks and preamplifiers. Also, we 
shall include any materials such as Airfoam 
rubber or Corprene, used for acoustic isolation 
in the case. We must consider the case from two 
separate points of view, electric and acoustic. 

The case is usually metal and is grounded by 
contact with sea water. We must therefore con¬ 
sider the capacitances of all terminals to the 
case, together with their associated dissipation 
factors. The magnitude of these effects depends 
upon the geometry and the materials which find 
themselves in electric fields, and in most well- 
designed transducers may be neglected; how¬ 
ever, it is important to discuss them in order 
that this desirable end may be achieved. 

In considering the elastic aspects of the case, 
we shall have to include any coupling fluid (e.g., 
castor oil). We must concern ourselves with 
cavity resonances within the oil, couplings from 
the motor to the case through various brackets 
and the oil, etc., and it is clear that the whole 
matter is extremely complicated. We have been 
unable to make any useful theoretical treat¬ 
ments of this problem, even of the crudest sort, 
to serve as a qualitative guide, and our results 
are fragmentary and primarily empirical. 

The ideal solution would be to make all cou¬ 
plings zero, except that from the primary face 
of the motor to the water via the window. Two 
distinct attempts at this have been made; the 
window-coupled gas-filled unit, in which the 
crystals see negligible impedance except on 
their primary radiating faces which look into 
water through a window usually made of rub¬ 
ber ; and the attempt to isolate acoustically all 
but the primary radiating faces of the crystal 
with Airfoam rubber, Corprene, etc. 

Isolation of the secondary faces of the crys¬ 
tals is important for at least three reasons: (1) 
to reduce dissipation of energy, (2) to prevent 
“holes” in the response caused by excitation of 
parasitic modes, and (3) to reduce cross talk 












DISSECTION OF A TRANSDUCER 


75 


if there are two motors in one case. No general 
solution of this problem has been found, but 
some instances are mentioned in Chapter 7. 


Bare Motor 

The bare motor consists of all the crystals 
together with any backing or fronting plates or 
bars to which they are attached. Electrically, it 
is a three-terminal network consisting of the 
two leads to the crystal and the backing plate, 
as shown in Figure 3. If the backing plate is 
glass or other nonconducting material, it is 
only a two-terminal network. 

With respect to the motor, we are in a more 
favorable position than in studying cases: sev¬ 
eral theoretical problems have been roughly 
solved; all of these are, of course, idealized, 
but they furnish valuable guidance to experi¬ 



mental investigation. Experimentally, we can 
make a very thorough examination of the 
motor, electrically with the three-terminal im¬ 
pedance bridge, and mechanically with the 
probe microphone (see Chapter 9). Results of 
these investigations are summarized later. 


Backing Plates or Bars 

We now begin a more detailed dissection, 
studying the parts of the bare motor itself; here 
we find systems of sufficient simplicity to enable 
quite accurate results to be obtained, both theo¬ 
retically and experimentally. Whatever its 
shape, a backing plate or bar is a well-defined 


isotropic elastic system and will possess a 
sharply defined spectrum of normal modes. If 
the geometry is sufficiently simple, we can cal¬ 
culate these normal modes with high precision 
and, in any event, we can map them and deter¬ 
mine their resonant frequencies with the aid of 
the probe microphone (see Section 3.4). 


Crystal Blocks 

In the assembly of motors, blocks of two or 
more crystals are often used and it is therefore 
important to know to what extent the proper¬ 
ties of such blocks differ from those of single 
crystals. In general, the results are just what 
would be expected; if the lateral dimensions of 
the assembled blocks are small compared to the 
longitudinal dimensions, the behavior closely 
follows that of a single crystal, assuming good 
cement joints. However, as soon as these dimen¬ 
sions approach equality with the longitudinal 
dimensions, undesired modes of motion begin 
to become important, and this sets an upper 
limit to the practical size of blocks. 


Single Crystal Plates 

We finally reach the primary active compo¬ 
nent of a transducer, the single crystal plate. 
This is susceptible to quite precise theoretical 
treatment and to detailed experimental exami¬ 
nation (see Section 3.2). 


Summary 

The results of our dissection may now be 
summarized as follows: Electrically, a com¬ 
plete sonar system is represented as shown in 
Figure 4. The circle with a question mark 
indicates the necessity of making some definite 
disposal of the transducer end of the shield; the 
dotted line indicates the most usual disposition, 
which is to leave it free or, in other words, to 
connect it to ground through a very small ca¬ 
pacitance. References are given to the quanti¬ 
tative discussions of each part of this circuit; 
for the present, we should merely observe that 










76 


COMPONENT PARTS 


the leads from the electronic system to the 
motor are shunted electrically in a number of 
ways. 

It should be emphasized that this figure is a 
correct representation only of the electric cir¬ 
cuit. The motor is connected mechanically to the 
case through its supports and any coupling fluid 
such as castor oil. If we knew how to replace 
Figure 4 by an equivalent circuit, these cou- 


itself, adequate for a satisfactory understand¬ 
ing of crystal transducers, because of the many 
other important effects; however, it is an essen¬ 
tial first step. 

In Chapter 2, the fundamental piezoelectric 
relations were derived and the boundary-value 
problem governing a loaded single rectangular 
crystal plate of finite width and thickness was 
solved to the next approximation beyond that 



Figure 4. Electric circuit for a complete transducer, with references to discussions of various parts. 


plings would appear as shunts across various 
elements in the more detailed equivalent circuit 
of the motor. In either case, the outgoing radia¬ 
tion is represented by the power consumed by 
a resistor inside the equivalent circuit of the 
motor, or, if the unit acts as a receiver, the in¬ 
coming radiation is represented by a generator 
in series with the above resistor. We now see 
that this resistor or generator is shunted by a 
maze of electrical and mechanical impedances. 

The balance of this chapter is devoted to a 
detailed study of the component parts qualita¬ 
tively discussed above. 


3 2 SINGLE CRYSTAL PLATES 

The most thorough understanding of the 
properties of single crystal plates is not, by 


given by Mason,^ by a semidirect variational 
method. The solution was left in the form of a 
one-dimensional boundary-value problem, iden¬ 
tical in structure to Mason’s, but having cor¬ 
rection terms added to the various parameters. 

In this section, the equivalent circuit corre¬ 
sponding to the solution of this perturbed 
boundary-value problem is deduced, some fea¬ 
tures of this circuit are analyzed, and the crys¬ 
tal constants significant to transducer design 
are evaluated. These constants are evaluated 
by experiments intimately related to the man¬ 
ner in which crystals are used in transducers, 
whereas some of the constants appearing in the 
literature depend upon a correct theoretical un¬ 
derstanding of high-order shear modes; in view 
of the complicated motions which are possible 
at high frequency, as shown by the silicon car¬ 
bide dust pictures in Figure 13 of Section 2.5.3, 


































SINGLE CRYSTAL PLATES 


77 


the utmost caution is necessary in interpreting 
measurements on the higher modes. A second¬ 
ary advantage of the procedure here used to 
evaluate the constants, from the limited view¬ 
point of this volume, is that one does not need 
to bother with a large number of constants 
which are of negligible importance in trans¬ 
ducer design. 


Ecjuivalent Circuit of a Loaded 
Rectangular Crystal 


In Chapter 2, it was shown that the motion of 
a rectangular plate of Rochelle salt [RS] or 
ammonium dihydrogen phosphate [ADP] is 
governed, to the next approximation beyond 
that given by Mason, by the one-dimensional 
boundary-value problem defined by equations 
(233), (234), (235), and (236) of that chapter. 

The solution of this boundary-value problem, 
with the two ends at 7/ = 0 and y — L, is 


V = Vi cos k'^y 4- 


{V 2 — Vi cos k'^L) sin k~^y 

sin k'^'L 


( 1 ) 


“ (il) [K'* *') }’ 


I ^ 

k = -, c 
c 


■ ar- 


(3) 


The boundary values Vi and Vo of equation (1) 
are determined by the boundary conditions 
equations (234) and (235) of Chapter 2; and 
the charge q on the positive electrode is obtained 
by integrating the charge density, equation 
(236). One thus has three linear equations 
from which q, Vi, and Vo can be determined if 
the potential difference A is known, 

Y+k+ ( - Yi CSC -f- ^2 cot d+) + N 2 V 2 

F+A 


t ’ 

Y+k+ ( — Vi cot 6+ -f- V 2 CSC e+) - NiVi 

_ 

t ’ 


(4) 


(5) 

( 6 ) 


e+ = k+L, C+ = ~ = C. (7) 

^Tvt 47 ^^ 

These equations are now transformed as fol¬ 
lows : the displacements are replaced by veloci¬ 
ties, according to I = q = mq, etc.; the stif- 
fances are replaced by the corresponding im¬ 
pedances, according to No = mzo, etc.; and fic¬ 
titious electric currents Ii and lo are introduced, 
proportional to the outward velocity of each end 
of the crystal, 

A = 0+( - yo, h = 4>^V2, (8) 

</)+=— F^w. (9) 

Finally, multiplying equations (4) and (5) by 
the radiation area wt and dividing by </>+, equa¬ 
tions (4), (5), and (6) become 


(/ - h - h) 

iwC 


= A, 


(ZI.ZO (fl). 


= A, 

(4') 

= A, 

(S') 


(60 

’ wt \ 
, 1 

(10) 


(11) 


Large Z, with various subscripts, is here 
used for equivalent electric impedances, and 
small z for specific acoustic impedances. The 
factor ivt/(f)+~ converts a specific acoustic im¬ 
pedance to an electric impedance. The character¬ 
istic impedance of the crystal is 


= pc I 1 - |”^g 2 ^ 4- efz- 


-d) 


{w + t) 
k-wt 


+ 




It is now readily verified that equations (4'), 
(5'), and (6') are the Maxwell equations for the 
three-mesh equivalent circuit shown in Figure 
5. Furthermore, this circuit is identical in 
structure to the Mason equivalent circuit, and 
the parameters differ only by correction terms 


q = C+A - F+w{V 2 - Fi), 














78 


COMPONENT PARTS 


which, although small, are of great practical 
importance. 

It should be noted that by shorting the con¬ 
denser in Figure 5, and using the proper pa¬ 
rameters, one has the equivalent circuit for a 



Figure 5. Equivalent circuit for a loaded rec¬ 
tangular crystal plate, to the next approxima¬ 
tion beyond that given by Mason. 

nonpiezoelectric plate, e.g., a backing rod. If 
the rod is driven by a force at either end, this 
force goes in series with Zj or Z^. 


^ Equivalent Circuit in Two Cases 
Definitions 

There are two special cases which are ex¬ 
tremely important, first, because they closely 
approximate the actual situation in three im¬ 
portant classes of transducers (backing plate; 
inertia drive and symmetrical drive), and sec¬ 
ond, because they are important in the experi¬ 
mental determination of crystal constants, the 
impedance of cemented joints, etc. These are 
defined as follows: 

Case I. Approximate Block. One of the 
terminating impedances (Z|) is large while 
the other {Zt) is small. A more precise mean¬ 
ing is given to “large” and “small” in the 
analysis which follows; for the present, it may 
be remarked that this condition is very well 
approximated over the practical operating band 
if one end of the crystal is cemented, by a good 
joint, to an approximately quarter-wave back¬ 
ing plate or bar, and the other looks into water 
or a lower impedance. Examples are University 
of California Division of War Research 
[UCDWR] types CQ, JB, GA, etc. 


Case 11. Approximate Free-Fi'ee. Both ter¬ 
minating impedances are “small.” This condi¬ 
tion is closely approximated in transducers of 
two classes: the inertia drive and the sym¬ 
metrical drive. 

Mechanical Arm 

The series equivalent of the two parallel arms 
is 

i iZt tan Y + j {iZ* tan ^ + Zt j 

( 2iZ+ tan y + Z| + Zj) 

Making the above approximations, one has for 
series equivalent of the two parallel arms 


Case I: 

(t + zt) - 


if 

1^5 -^2 i 1 Z| 1 , 

(14) 

Case II: 

ht + i(Z| -f zt) 

(15) 

if 

IZti, IZtl « \t\, 


in which t 

is used as a temporary abbreviation 


for tan {6^/2). 

Combining this with the esc <9+ term, one has 
the circuit shown in Figure 6, which has only 
one mechanical arm. The values of the imped¬ 
ance of the mechanical arm in the two cases, 
together with two trigonometric identities used 
to obtain these results, are 

0 + 

— CSC d+ -f tan -^ = - cot 0+, 

0 + 0 + 

- CSC 0+ + 5 tan \ cot 

Case I: Z| = - iZt cot 0+ + Zt 

(iZt tan ^ + Zty 
-zi-’ 

Case II: Z|=-tiZtcot^-t-|(Z| + Zt). (17) 

These approximations are valid if the in¬ 
equalities of equations (14) and (15) are satis¬ 
fied ; therefore, to determine the significance of 
these inequalities, we need to find the order of 
magnitude of Z+ tan {6^/2) over the practical 
operating band, in each of the two cases. For the 
purposes of this rough estimate, it is permis- 























SINGLE CRYSTAL PLATES 


79 


sible to ignore the imaginary part of which 
is small; whereupon the mechanical or constant- 
voltage resonance (frequency of minimum im¬ 
pedance of the mechanical arm) occurs near 
6 = jt/2, 3 .t:/ 2, etc. in Case I and near 6 = n, 
3jt, etc., in Case II. These values make tan 
^V2 = 1 in Case I and infinity in Case II, both 
results being favorable to satisfying the in¬ 
equalities. Thus, very near to the respective 



A 



zt OR zt 


o——— 






B 


Figure 6. Equivalent circuit for Cases I and II. 

A. Circuit with cosecant term separated out. 

B. Circuit with cosecant term combined as given 
by equations (16) and (17). 

resonances, the inequality for Case I requires 
that |Z+1 and '\Zt\ be small compared to 
\Z\\, while that for Case II is satisfied by any 
finite Z\ and Zt. 

However, we need to know the conditions for 
validity of these inequalities over the practical 
operating band, which is determined largely by 
the Q of the mechanical arm. First, consider 
only the lowest resonance in each case. Then, in 
Case I the inequality will be improved by going 
to lower frequency; going to higher frequency, 
tan (i9+/2) will have become only 2.4 at a fre¬ 
quency 1.5 times resonance, which is much 
wider than the band width determined by the 
Q of the mechanical arm (see Section 3.2.2, 
Constant-Voltage Band Width). Turning to 
Case II, we see that if the frequency is as low as 
0.5, or as large as 1.5, times the resonant value, 
then tan (6»V2) will have fallen only to unity. 

Thus, in a band centered at the respective 
first resonances and wider than the practical 
operating band, the above inequalities will be 


valid and, with them, the approximate imped¬ 
ances given in equations (16) and (17), if 

Case I: |Z+|, [ i « \Z\\, (14') 

Case II: i Z| |, | Zt | « [ Z+j. (15') 

A more careful analysis shows that these con¬ 
ditions are also sufficient for the validity of the 
above approximations at the higher resonances; 
for, although the band width of validity de¬ 
creases as we go to higher resonances, the band 
width of practical operation also decreases. 

The characteristic impedances of water or 
rubber, ADP or RS crystals, and steel are ap¬ 
proximately in the ratio 1 to 4 to 26. Hence, we 
see that in Case I, even an eighth-wave steel 
backing bar or plate is such an excellent block 
that we make negligible error by dropping even 
the first-order correction term in equation (16), 
provided the cemented joint has a high im¬ 
pedance. Conversely, this correction term can 
be used to evaluate the impedance of cemented 
joints by observing the discrepancy between the 
resonance frequency and that for a perfect 
clamp (see Section 3.3) ; in this case, the other 
end of the crystal is made to look into air 
(Zt = 0), and the numerator of the correction 
term, evaluated at resonance, becomes simply 
—Z+|. In Case H, if Z|, and Zt are the im¬ 
pedances of water, then the inequality equation 
(15') is well satisfied so long as we save the 
first-order correction term i/4 (Z| -|- Zt). 

Resistance and Reactance 

We can now reach suggestive tentative con¬ 
clusions concerning the internal dissipation, 
since this can be evaluated except for two (fre¬ 
quency-dependent) parameters, the imaginary 
parts of T and N ; this leaves us in ignorance of 
the magnitude of the effect and how this varies 
with frequency, but enables us to determine its 
dependence on other quantities. 

The internal dissipation resistance is the real 
part of the cotangent terms in equations (16) 
and (17). Both Zt and 0^ have small imagi¬ 
nary parts which are readily separated by using 
the addition theorem for the cotangent. Since 
the imaginary parts of Zt and (9+ are both 
small, we see that they contribute only a second- 
order correction to the intrinsic reactance; 

















80 


COMPONENT PARTS 


since other second-order terms have been 
dropped, this one must also be dropped to obtain 
a consistent approximation and hence the in¬ 
trinsic reactance is obtained by replacing Zt 
and 0+ by their real parts in the two cotangent 
terms. The intrinsic resistance, however, gets 
a first-order term from both and (9+. The 
1/Z| term in Case I is also dropped, since this 
term is needed in only two instances: (1) the 
cement joint does not have high impedance, a 
situation which can and should be avoided by 
good technique (see Section 3.3), or (2) to 
evaluate the impedance of cemented joints, in 
which case precautions are taken to make T 
and N negligibly small. 

The impedances given by equations (16) and 
(17), with the intrinsic terms separated into 
their reactive and resistive parts and the 1/Z\ 
term dropped, then become 

Case I: 

ZX = — i{(RZ^) cot ((R0“^) -|- “b Z' 2 - (16^) 

Case II: 

+ m (17-, 

Rt = Rt' CSC- ((R0+) -b i?fG^((R0+), (18) 

R} = IRt csc= ( ^ + RtG,, ( ^2“ )' (19) 

Gn(cx) =(-11 CSC- a -b j cot a, (22) 

in which (R denotes the real, 3 the imaginary, 
part of the indicated quantity. The sign ( + ) 
on the resistances Rj; and Rt arises solely 
from the occurrence of the factor 
which converts mechanical impedance to equiv¬ 
alent electric impedance. 

It is readily shown that the squared cosecant 
and Gy functions vary negligibly over the prac¬ 
tical operating bands, whether these are 
centered at the first or higher resonances, and 


hence equations (18) and (19) may be evalu¬ 
ated at resonances for all practical purposes, 

Rt = Rif + n^\yRi, (18') 

Even without knowing the values of :sT and 
aN', some interesting tentative conclusions can 
be reached by studying equations (18') and 
(19'). First, let us see if they make reasonable 
predictions in instances where the results can 
be checked by simple considerations. 

A symmetrically loaded crystal may be re¬ 
garded as two approximately blocked crystals 
in parallel, since each half acts as a quarter- 
wave backing bar for the other half. An inertia 
drive can be similarly regarded, except that the 
radiation resistance is halved. Therefore, the 
predicted ratio of loss to radiation resistance 
for a blocked and a symmetrically loaded crystal 
should be the same, and an inertia drive should 
be half this. In verifying this, one must notice 
that the shape factors, based on actual dimen¬ 
sions, will be twice as great in the blocked as in 
the symmetrically loaded or inertia drive case; 
thus the dependence of Rt and Rti on these 
shape factors must be taken into account. We 
therefore have for the ratio of loss to useful re¬ 
sistance. 


blocked: 


{Rtb -b 




R\b) 


R2 


(23) 


symmetrically loaded: 


(f) 


= Equation (23) 


(24) 


inertia drive: 


^ R.vb 

2 


ti) 


= Twice equation (23), 
(25) 

and we see from equations (23), (24), and 
(25) that the theory gives sensible results in 
this case. The signs ( + ) have been dropped in 
equations (23), (24), and (25), since in form- 











SINGLE CRYSTAL PLATES 


81 


ing the ratios, the transformation factor 
( 2 vt/(f)+-) falls out and one is left with the cor¬ 
responding ratios of mechanical resistances. 

Next, consider the dependence upon n. For 
any one type of drive, consider the problem of 
radiating energy at a given frequency, so that 
T and N will be unchanged, but comparing the 
efficiency of doing this with the first (n = 1) 
and the second (n = 3) resonances. Keeping the 
frequency constant will require that the crystals 
working at their second resonance be about one- 
third as long as those working at their first 
resonance; however, this reduction will alter 
the radiating area in the same proportion as 
the lateral area, and hence we conclude, from 
equations (18') and (19'), that the ratio of 
tangential loss to useful radiation will be the 
same for 7i = 3 as for n = 1, while the ratio of 
normal loss to useful radiation will be about 
nine times as great. 

This is physically reasonable, and hence gives 
further confidence in the essential correctness 
of the variational treatment, as may be seen 
as follows: Consider first the normal loss. The 
dimensionless Poisson “breathing” ratios are 
easily shown to be (^(;/L)£„.(9^- and (^/L)8,(9+, 
constant except for the factor 6+. Hence, the loss 
must go up like 0+-, that is, like n-. One might, 
of course, try to terminate the lateral faces with 
an N so small as to make this loss still negligi¬ 
ble, but certainly there is nothing gained, from 
the viewpoint of efficiency, in using higher 
resonances. 

The tangential motion, although more com¬ 
plicated the higher the resonance used, depends 
on the average value of sin^ ((R0+), always 
over an integer number of quarter cycles, and 
hence should be independent of n, as it is from 
the theory. 

With regard to the dependence of Rt and 
that is 3T and 3N, on frequency, and ar¬ 
rangement of crystals and other obstacles inside 
a transducer, only some qualitative conjectures 
can at present be made, since there is very little 
experimental data available and the problem is 
too complicated to obtain a reliable theoretical 
estimate. 

The tangential impedance arises from the 
generation of viscous shear waves and these 
have an extremely short wavelength and are 


absorbed in a very short distance (of the order 
of a fraction of a millimeter in castor oil at 
room temperature). Accordingly, we may ex¬ 
pect to get some very reliable estimates by con¬ 
sidering an infinite plane vibrating tangentially 
and looking into a fixed plane at distance D. 
This situation is discussed in Section 2.1.8 and 
one concludes that if D is greater than about 
one millimeter, the effect is probably unim¬ 
portant, but that it may become extremely im¬ 
portant if crystals with relative tangential mo¬ 
tion are separated by a smaller distance. An 
isolated but very impressive experimental re¬ 
sult supports this viewpoint in the case of a 
modified LFCDWR-type CY4 transducer.*^ The 
tangential impedance T is independent of fre¬ 
quency for small D and varies as for large 
D. This latter function of co changes by less 
than 10 per cent over the practical operating 
band, and hence we conclude that the frequency 
dependence of Rt is probably not very im¬ 
portant. However, the temperature dependence 
of Rt would be expected to be strong, since 
it is proportional to the square root of the vis¬ 
cosity for large D and to the viscosity itself for 
small D. 

The variation of Rt is much more difficult 
to estimate. It presumably depends upon the 
standing-wave pattern established inside a 
transducer, which can vary with extreme ra¬ 
pidity as the frequency varies, together with 
the dissipation mechanisms which absorb 
energy from these standing waves. A very 
crude semiqualitative discussion of this very 
complicated problem is given elsewhere,- but it 
is doubtful that one will come to any practically 
useful conclusions in this matter without 
further experimental study. 

Mechanical Resonance and Antiresonance 

The mechanical resonance frequencies for the 
two cases can be estimated as follows. Assum¬ 
ing that Zt in Case I, and both Z\ and Z\ in 
Case H, are purely resistive, the resonant fre¬ 
quencies are given by 


Case I: 

„ , TItt 

= y, 

(26) 

Case II: 

(R0+ = n-K, 

(27) 


in which 7i — 1, 3, 5, etc., for the first, second. 


See Section 7.4. 




82 


COMPONENT PARTS 


third, etc., resonance in both cases. Now, neg¬ 
lecting any reactive part in T and N, one has 

= [ 1 + {eiri + 4ri) ], 

in which and are the ratios of width to 
length and of thickness to length, respectively. 
Putting the 0th approximation for kL into the 
correction term, and inverting, one has a gen¬ 
eralization, to the anisotropic case, of the Ray¬ 
leigh frequency formula 

Case !• 

Case II: 

= + (30) 

in which the relation k = 2nf/c has been used. 
It should be noticed that equation (29) can be 
deduced from equation (30) by thinking of the 
blocked crystal as a free-free crystal of twice 
the length; then this equation reads 

C 3 S 0 I* 

fr{2L) = 2 ) { 1 “ ( ^ ) (29') 

It is seen that the effect of the lateral inertia 
correction increases with increasing n, in such 
a way that the successive resonances fall far¬ 
ther and farther below the simple 1, 3, 5, etc., 
ratios predicted without this term. Quantita¬ 
tive verification of this is given in the next 
section. 

The mechanical antiresonant frequencies are 
given by equations (29) and (30) by setting 
n = 0, 2, 4, etc. These are extremely difficult to 
observe in the laboratory since the condenser 
is shunted by a very high mechanical imped¬ 
ance: one needs a very delicate probe micro¬ 
phone; however, they are readily observed in 
the responses of complete transducers (see 
Chapter 4). 

It is scarcely necessary to emphasize that 
these equations are valid only if the diminuend 


in the correction bracket is small compared to 
1; however, this covers all practical cases. 

These equations are used in the next section 
to determine the crystal constants c, e,^., and 
E^; from the first of these and the density, one 
obtains the characteristic impedance for a nar¬ 
row and thin plate. 

Constant-Voltage Band Width 

The band width for coyistant-voltage drive 
can now be estimated from equations (16') and 
(17') ; this is an important design parameter, 
but it should be emphasized that it is not the 
actual band width realized in practical trans¬ 
ducers, since this latter depends upon the 
matching network and amplifier used. In ac¬ 
cordance with general usage, the half-iuidth is 
defined as the change in frequency bf required 
to reduce the power dissipated in the radiation 
resistance to half its resonance value, and the 
band width is twice this quantity. In this calcu¬ 
lation, it is assumed that the radiation imped¬ 
ances [Zt in Case I, 14 (^1 + ^t) in Case II] 
are purely resistive, an assumption which is 
justified by observing that any reactive part, 
if varying slowly, can be combined with the in¬ 
trinsic reactance, the primary effect being to 
shift the resonance frequency slightly. 

The half-breadth in (R6+ is first calculated 
from equation (16') and (17'). For this pur¬ 
pose, it should be noticed that depends upon 
(R 61 + only in its correction term, and this latter 
is therefore evaluated at resonance. Further¬ 
more, one keeps only the first-order departure 
of the reactance from its zero value at reso¬ 
nance, verifying the validity of this and the 
previous approximation a posteriori. The half¬ 
breadth in (R^+ is then determined by the con¬ 
dition that the reactance is equal to the re¬ 
sistance. 

Case I: 

((RZtcsc2 (Rd+)rd(Rd+ = Rt Rt, (31) 

Case II: 

}3lZ+csc=(-^ + +^2+). (32) 

Now (6(R^+/cR(9+)r = hf/fr since the correc- 





SINGLE CRYSTAL PLATES 


83 


tion factors to evaluated at resonance, 

cancel. Therefore 


Case I: 


/25/\ / 1 \ 

/ 4 \ (-i- Rt) 

i fr )~\Qm) 

Utt/ ((HZ+), 

Case II; 


(26/) / 1 ) 

1 

fr \Qm] 

1 

{±\ 


\nirj 

((RZt). 


In the above ratios of equivalent electrical 
quantities, the conversion factor wt/cf>+- cancels 
out, leaving the ratio of mechanical quantities; 
however, the dependence of on width re¬ 
mains, and increases the band width somewhat. 

From equations (33) and (34) one sees that 
a blocked and symmetrically loaded crystal has 
the same fractional band width, about one-third 
for a high-efficiency unit, increasing with in¬ 
ternal losses; an inertia drive crystal (i?t = 0) 
has about half this fractional band width, also 
increasing with losses. The corresponding me¬ 
chanical Q’s are 3 or less and 6 or less, respec¬ 
tively. One other important conclusion is that 
the fractional band width decreases as the 
order of the resonance increases, being only 
about one-third as large at the second resonance 
as at the first, etc. This result is readily observ¬ 
able, and for most applications is a serious ob¬ 
jection to using higher resonances. 

Electrical Antiresonance ; 

Transformation Ratio 

Referring to Figure 6, one sees that at some 
frequency above the mechanical resonance, the 
mechanical arm will become inductive by an 
amount sufficient to form a resonant loop with 
the condenser. At this frequency, f^, the total 
series equivalent impedance seen at the electric 
terminals will be large (infinite in a lossless 
system), so that this loop resonance will appear 
at the electric terminals as an electric anti¬ 
resonance. 

This effect is considerably obscured if there 
is a resistance as large as that corresponding 
to the crystal working into water, but in an 
efficient transducer will appear as a wiggle in 


the series equivalent reactance (see Section 
3.2.2). For a free-free crystal in air, the effect 
is very marked, and the observed difference be¬ 
tween the mechanical resonance and the elec¬ 
tric antiresonance enables the transformation 
ratio between electric and acoustic impedances, 
(^+-, and hence the piezoelectric constant F+, to 
be determined. Only this case will be discussed 
here. 

The total admittance of the free-free crystal 
in air, in which case Zf and 6^ are real, is 

icoC + (^) tan (35) 

and this is zero at the electric antiresonance. 
Evaluating all quantities in equation (35) at 
resonance, except the rapidly varying tangent 
term, one has 


COrC — 


4 

{Zt)r {et - Ot) 


= 0 , 


(36) 


(et - 0t) _ {fa - fr) _4 


d+ 

(fa - fr)L 


fr (wZ^c) rCniT 

^ 2L{4>+^-)r 

mr~Cwt(zt)r’ 

nTrK(zt)r' 


, (37) 

(38) 


The quantities (F+-),- and {Zt)r depend 
upon 71 and the shape factors iv/L and (t/L). 
Neglecting this variation for the moment, that 
is, considering only very narrow crystals, we 
see that the theory predicts that the product 
(fa — fr)L for any fixed 7i will be constant for 
crystals of all lengths, and that its values for 
the successive antiresonances are in the ratio 
1, Vs, Vs, etc. 

These and the more general predictions of 
equation (38) are compared with experiment in 
Section 3.2.3, where some but not all these pre¬ 
dictions are verified. The discrepancy occurs 
primarily in the predictions at the higher anti¬ 
resonances and since, as has previously been 
remarked, there are several reasons why it is 
unsatisfactory to operate a transducer very far 
from its first resonance, it is believed that this 
failure of the theory is not important from the 
viewpoint of this volume. A possible explana¬ 
tion of this partial failure of the theory, in the 
face of its close agreement with more signifi- 














84 


COMPONENT PARTS 


cant experimental results, is offered in Section 

3.2.3. 

Low-Frequency Limit 

At a sufficiently low frequency, a crystal be¬ 
comes stiffness controlled. This limiting case is 
the basis of the experimental determination of 
the dielectric constant A 33 and also shows very 
simply why the open-circuit response of a hy¬ 
drophone approaches a constant as the fre¬ 
quency is reduced. 

In this limit, far below resonance, the re¬ 
actances become so large that all resistances 
may be neglected. A further simplification is 
that the finite width corrections become negligi¬ 
ble, so that 6^ becomes simply o)L/c; all signs 
( + ) may therefore be dropped, i.e., the Mason 
approximation is valid. 

The radiation reactances, Z 2 in Case I, Zi and 
Zo in Case II, also become negligible, even in 
comparison with the tan (9/2) terms, as will 
now be shown. The tan (9/2) term becomes 
approximately 

m)(?) - '(f )-(?)■ ®> 

which is the reactance of the mass of half the 
crystal. The radiation reactance is roughly that 
of the mass of a parallelepiped constructed on 
the radiating face and extending outward a dis¬ 
tance of the order of the smaller of the two 
dimensions of this face t. The volume of this 
parallelepiped is 2 vt-, and hence the radiation 
reactance is smaller than that arising from the 
mass of the crystal by a factor of order t/L. 
The radiation resistance is smaller by another 
factor of order (t/L). 

In spite of the stiffness of the whole crystal, 
there is still a definite meaning attached to 
blocking one end. Case I: the number 1 end is 
blocked if L is small, compared to h. If Zi 
arises from a backing bar of length L^, then its 
reactance is approximately io^Lf^co( wt/<j>-), and 
the ratio of the currents is 



Since the density of backing material, usually 
steel, is greater than that of the crystal, any 


ordinary backing plate will make the above 
ratio small compared to 1 ; this is the definition 
of Case I. In Case II, since both Z^ and Zo are 
negligible compared to the tan (9/2) terms, this 
current ratio is 1 . 

We therefore have the cosecant stiffness term 
in series with (Case I) iZ^ tan (9/2) or (Case 
II) (i/2)Z^ tan (9/2). The impedances of these 
combinations are 


Case I: 


— iZc cot 9 = 1 


(1 

). (41) 

Case II: 




). (42) 


At sufficiently low frequency these differ negli¬ 
gibly, and in the limit the mechanical arm be- 



Figure 7. Equivalent circuit for Cases I and 
II, in the low-frequency limit. 


comes a constant condenser of capacitance 
(L/cZJ = (47iF-/qc-) (wL/4nt) in both cases, 
so that the effect is to increase the purely elec¬ 
tric dielectric constant by (AitF-Zgc-). The 
circuit is shown in Figure 7. 

The fractional error in the reactance of the 
mechanical arm is of order 9-/S and (0/2) “/3 
in Cases I and II, respectively, and since the 
resonance frequency corresponds to 9 = jt/2 
and 9 = 71, respectively, we see that the per¬ 
centage error is the same in the two cases at 
any given fraction of the respective resonance 
frequencies. This error is about 20 per cent at 
half, and less than 1 per cent at one-tenth, of 
the resonance frequency. The error in the series- 
equivalent reactance is considerably smaller, 
since the mechanical condenser is much smaller 
than the electrical. 










SINGLE CRYSTAL PLATES 


85 


The ratio of the mechanical to the electrical 
condenser, for Case I, is 

and the right member of equation (43) is also 
valid for Case II, since it is expressed in terms 
of the ratio of the actual to the resonant fre¬ 
quency. The dimensionless crystal constant 
4:7cF^/Kqc^ is shown in Section 3.2.3 to be of the 
order of a tenth for both RS and ADP, and 
hence a small correction must be applied to 
capacitance measurements even at very low 
frequencies to get the blocked dielectric con¬ 
stant. This dielectric constant is evaluated in 
Section 3.2.3. 

The equivalent circuit of a hydrophone, in 
the low-frequency limit, is obtained from Fig¬ 
ure 7 by inserting a generator in the mechan¬ 
ical arm. The open-circuit emf of this generator 
is proportional to the pressure amplitude of the 
incident sound wave and, in the low-frequency 
limit, its internal impedance is negligible. The 
circuit then becomes a condenser potential- 
divider and the open-circuit voltage across the 
electric condenser is therefore independent of 
frequency. The voltage-transfer factor is ap¬ 
proximately equal to the capacitance ratio, 
equation (43). However, what one wants is the 
transfer factor which converts pressure to open- 
circuit voltage, and this involves another so 
that one ultimately finds that the open-circuit 
response, in volts per unit incident pressure, is 
proportional to F as one would expect, rather 
than to F~. The details of these matters are 
discussed in Chapter 4, the present brief dis¬ 
cussion being intended only to show one appli¬ 
cation of the circuit for the low-frequency limit. 


Series-Equivalent Impedance 


The series-equivalent impedance of the paral¬ 
lel electrical and mechanical arms can be meas¬ 
ured with precision by a suitable bridge. The 
value deduced from the equivalent circuit is 


^ (-i/o:C) (R + iX ) 

' lR^i{X-l/<^C)Y 


(44) 


in which R here includes both loss and radiation 
resistance, X is the total equivalent reactance 


of the mechanical arm, and the signs ( + ) are 
to be understood. These two quantities are de¬ 
fined by equations (16') and (17') for the two 
cases. 

The real and imaginary parts of this imped¬ 
ance are 



D{io) 


1 


1 ^ ^ 

r(x - i/coC)“i 

\ c^CR 1 

L J 


(X - i/coon -^ 
R2 J : 


(45) 

(46) 

(47) 


in which Z) (co) is the distortion factor by which 
the impedance of the condenser must be multi¬ 
plied to obtain the actual series-equivalent re¬ 
actance. 

The factors l/oj-C^R and l/atCR are slowly 
varying compared to (X — 1/ooC), and hence 
these will first be regarded as constants. The 
error arising from this assumption is not great, 
and will be eliminated later. 

The quantity (X — 1/coC) is the reactance 
around the mesh formed by the electrical and 
mechanical arms, and therefore vanishes at 
the resonance frequency of this mesh, that is, 
at the electrical antiresonance. Neglecting any 
radiation reactance, it is 


Case I: X - ^ = (5lZ+ cot ye - (jL), (48) 

CaseII:X-i=-i<RZtcot|-(J-J, (49) 

in which y = ((R0+/0),. is the reciprocal of the 
frequency-depression factor, at resonance; it is 
evaluated at resonance because it differs from 
1 only by small quantities whose variation over 
the operating band is a second-order small 
quantity. 

Expanding the mesh reactance near the anti¬ 
resonance frequency, one can readily show that 
only the first-order term is of any importance 
and one therefore has for 


Case I: 


X - 1/coC 

R 


y(tiZ+ 

R 


(d - da). 


(50) 


Ignoring the slight departure of the /,.//„ 
from 1 in the coefficient, this is just 













86 


COMPONENT PARTS 


the same result holds in 
Case II provided the proper is used. 

Putting equation (50) into equations (45), 
(46), and (47), with the slowly varying quan¬ 
tities evaluated at resonance, one has 


Rs — i2max(l + Xr) h 
p_ L 


(51) 

(52j 


frequencies, decreasing with increasing R, and 
has the same fractional band width as the con¬ 
stant voltage response, both being 1/Q,;. The 
distortion factor Zl(a)) indicates that the series- 
equivalent reactance approaches the condenser 
value between the resonances, being less just 
below /„ and greater just above. 

Recalling that the slowly varying quantities 
l/oiCR and l/(a~C~R were evaluated at reso- 



FREQUENCY IN KC 

Figure 8. f-R, and (1/C) for CCUlOZ-1 transducer (0.773" motor). 


■0(») = 1 +(-^)^(l (53) 


These results are valid for both cases, provided 
the proper Q’s are used. The quantity is 
(jdrCR and, neglecting finite width corrections, is 
related to by 

QmQe = (55) 

The series-equivalent resistance has a maxi¬ 
mum value of (1/co^^C-R) at the antiresonance 


nance, we now see that the actual variation of 
these quantities will skew the R and B curves, 
slightly moving the extrema toward lower fre¬ 
quencies, lifting the parts of the curves below, 
and lowering the parts above, /^. 

For practical design purposes, the foregoing 
treatment is adequate, but in checking the theory, 
it is convenient to plot f-R^ and (—coXJ, the 
latter being the reciprocal of the series- 
equivalent capacitance as read directly from the 
bridge. According to the theory, these should 
be 1/(1 + x-^) and [1 (1/Q^)x/(1 + x^)], 
























































































SINGLE CRYSTAL PLATES 


87 


except for a scale factor. These quantities 
are plotted for two transducers in Figures 8 and 
9. Three features worthy of note are the symme¬ 
try of the curves, the occurrence of the maxi¬ 
mum f-R^ at the midpoint of (—coXJ and the 
occurrence of the extrema of (—coX^) at very 
nearly the same frequencies as the half-value 
frequencies of pR^. 


having the symmetry of RS or any higher sym¬ 
metry. 

The performance of a transducer at higher 
resonances is inferior in many respects to its 
performance at the first resonance; therefore, 
while this formula for n = 1 has very im¬ 
portant practical application to design, its pre¬ 
dictions for larger 7i’s are not directly signifi- 



FREQUENCY IN KC 

Figured. f-R, and (1/C) for CCUlOZ-2 transducer (high-frequency motor). 


Evaluation of Crystal Constants 


Young’s Modulus, Characteristic Impedance 
AND Width Poisson Ratio 

From equation (30) of Section 3.2.2, the res¬ 
onance frequencies of a free-free crystal are 
given by 


n 


+ ( 56 ) 


in which n is the modal number, with values 
1, 3, 5, etc. This is the generalization of the 
Rayleigh formula for the resonance frequencies 
of a rod of finite width, for an anisotropic rod 


cant. However, it is extremely important to test 
the validity of the general theory in as many 
ways as possible, and the various order reso¬ 
nances offer an excellent opportunity for this. 

Accordingly, the resonance frequencies of 
about 60 45° Z-cut ADP rectangular plates, 
with evaporated gold electrodes, have been 
measured. The results are given in Figure 10, 
in which f,L/n is plotted against rp. Each point 
represents a group of 2 to 6 crystals having the 
same dimensions. The frequencies were meas¬ 
ured with a Western Electric 17B oscillator 
which had been checked against a Bendix CRR 
heterodyne frequency meter, and other precau- 





















































































«5 

60 

55 

50 

45 

40 

35 

30 

25 

20 

10 . 


COMPONENT PARTS 



SQUARE OF WIDTH-LENGTH RATIO 


ipendence of 


resonance frequencies of ADP on 


length, width-length ratio, 


and modal 



















































SINGLE CRYSTAL PLATES 


89 


tions were taken to reduce error. The 1-db 
breadth of the resonances was less than 15 c for 
all crystals. The dispersion of the points from 
the linear part of each curve, which presumably 
includes the effect of all random errors, is about 
0.5 per cent. 

The thickness-length ratio varied, the largest 
value being 0.2. The effect of thickness on the 
resonance frequencies was less than the experi¬ 
mental dispersion from a smooth curve; we 
therefore conclude that the thickness Poisson 
motion is of negligible importance for the range 
of thickness-length ratios used in practical 
transducers. This conclusion is supported by 
results given in Section 3.3, where a rough esti¬ 
mate of 0.2 is obtained for e,. This very rough 
value indicates that the thickness correction 
term is 

(fj) ~a X 10-*) n'-. (57) 

Thus, the lack of any measurable dependence 
for n = 1 and 3 is readily understood. For 
n = 5 and 7, however, the fractional effect of 
the largest thickness ratio 0.2 should be about 
2 and 4 per cent if the above value of e, is cor¬ 
rect, whereas no such effect was observed. Two 
possible explanations of this suggest them¬ 
selves : first, E, may be smaller than the above 
rough value and, second, the error in these 
measurements may be greater for n = 5 and 7. 
This latter view is supported by the greater 
irregularity of the points on the curves for the 
higher n’s. In any event, this is a matter of 
higher precision rather than general principle, 
and certainly the thickness effect is not of any 
great practical importance. 

The general structure of these curves is in 
excellent agreement with equation (56), out to 
values of 7’,^- where the fractional correction be¬ 
comes of order 1 / 3 . Beyond this, higher-order 
terms become important enough to introduce 
curvature. It should be emphasized that this 
curving in no way casts doubt upon the validity 
of the variational principle from which the per¬ 
turbed equivalent circuit was deduced, but is 
an anticipated consequence of the approxima¬ 
tions made in applying that principle, since 
terms in the square of the fractional correction 
were dropped. Furthermore, it should be noticed 


that for n = 1, the only n of practical impor¬ 
tance in design, the departure from a straight 
line is negligible. 

According to equation (56), the intercept of 
these curves is c/2, which is about 64.7 kc-in. 
We therefore obtain 

ADP: c = 3.29 X 10^ cm/sec. (58) 

Combining this with the density of ADP, 1.80 g 
per cu cm, one has for the characteristic im¬ 
pedance at zero width 

ADP: Zc = pc = 5.92 X 10® g/cm^ sec. (59) 

Equation (56) predicts that the initial slopes 
of these lines, neglecting the thickness term, are 



In Table 1, the measured slopes, and the slopes 
divided by 71- are given. In view of the great 
range of the slope, the constancy of the last 
column is regarded as good. 


Table 1. Slopes and slopes/w^, from Figure 10. 


n 

Slope, kc-in. 

(Slope/w2), kc-in. 

1 

12.9 

12.9 

3 

125.0 

12.6 

5 

325.0 

13.0 

7 

630.0 

12.9 

9 

1070.0 

13.2 



Average 12.9 ± 0.3 extreme 


From equation (60), together with the fig¬ 
ures in Table 1. one obtains 

ADP: eu, = 0.7. (61) 

We know of no data on the higher resonances 
of free-free 45° Z-cut ADP plates to compare 
the foregoing results, but BTL^ and NRL^ have 
given data‘s on the first resonance. 

NRL makes no mention of any thickness 
effect, and the width effect agrees quite well 
with our data, as shown by triangles on the 
line for tz = 1 in Figure 10. 

BTL finds a small thickness effect, as indi¬ 
cated by Figure 11; however, the effect for 

<= This work contains valuable material on transducer 
design, but the approach is so ditferent from ours that 
we have not been able to study it thoroughly in the 
time available. It would undoubtedly repay a careful 
study. 








90 


COMPONENT PARTS 


Vf = 0.2, the largest value used in obtaining 
Figure 10, is about equal to the dispersion of 
the points and therefore, as previously re¬ 
marked, our data is not sufficiently accurate to 
resolve the effect. By plotting the intercepts of 
Figure 11, one gets a value of about 0.3 for e,; 
this is higher than the rough estimate of 0.2 
previously mentioned, but one would need to 
study the original data to decide if the differ¬ 
ence is significant. A curious feature of BTL’s 


value of £,^,(0.7) is over 4 times as large as an 
early value (0.17) given by Mason.^ Mason has 
since indicated in conversation that the value 
given in the quoted memo is low, but has not 
indicated by how much. 

From the standpoint of determining the res¬ 
onance frequency of 45° Z-cut plates for design 
purposes, the discrepancy is unimportant be¬ 
cause, regardless of interpretation, the data 
given in Figure 10 can be used empirically. 



.1 .2 .3 .4 .5 ,6 ,7 .8 .9 1,0 

RATIO WIDTH TO LENGTH (^) 


Figure 11. 45° Z-cut ADP crystals, product of resonance frequency and length against ratio of width 

to length for various ratios of thickness to length. (From BTL Drawing BA262699.) 


curves is their congruence for all Vf as ap¬ 
proaches 1. This implies that the resonance 
frequencies of a thin square plate and a cube 
are the same, a result which is difficult to under¬ 
stand. 

BTL’s data for = 0.1 agree quite well with 
ours, as indicated by the squares in Figure 10. 

Despite the good agreement of the frequency 
depression data for the first resonance, our 


However, it is extremely important that the 
theory be in good order, and we therefore feel 
that the original data, from which the quoted 
memo was prepared, should be carefully re¬ 
studied. It seems possible that the enormous 
discrepancy may arise in some way from the 
indirect method used to determine e,^., involving 
several different cuts and shear modes with 
resonance frequencies up to a megacycle; how- 













































SINGLE CRYSTAL PLATES 


91 


ever, it is then hard to understand why there 
is such close agreement on the Young’s modulus. 
In a field as complicated as this, and thinking 
only in terms of the restricted purpose of this 
volume, it is probably a good policy to make 
crystal measurements directly on 45° Z-cut 
plates so as to obtain data intimately related 
to the manner in which the crystals are used 
in underwater transducers. If a particular ef¬ 
fect cannot be measured with 45° Z-cut plates, 
then it is of no consequence to underwater 
transducers. 

On the basis of the foregoing, we conclude 
that at least the first five consecutive resonances 
of a 45° Z-cut ADP plate correspond to longi¬ 
tudinal modes with lateral Poisson “breathing” 
coupled into them, provided the width-length 
ratio is not too large. The frequencies of these 
resonances are given by 

frL = ^(64.7 - 12.9nVf.), 

= n(64.7) (1 - 0.20/1^2). ^ ^ 

This result is valid near room temperature, and 
the temperature correction is so small as to be 
of negligible practical importance. 

As the width-length ratio increases, the lat¬ 
eral motion ultimately becomes so large that 
the modes became more complicated, as pre¬ 
dicted by the theory and as observed experi¬ 
mentally (see Figure 12 of Section 2.5.3). How¬ 
ever, it is not worth while to try to treat these 
more complicated modes theoretically because 
in practical transducers the width-length ratio 
should always be kept small enough so that 
these complicated motions do not occur. 

The modes of RS undoubtedly follow a pat¬ 
tern similar to ADP, but to the best of our 
knowledge have not been studied in detail. One 
will expect the more complicated type of motion 
to appear at a lower width-length ratio because 
of the somewhat larger and the shear of the 
electrode faces into a parallelogram (see Fig¬ 
ures 12 and 14A, 14B, and 14C of Section 2.5.3 
and the accompanying discussion where it is 
shown that the shear Poisson ratio e, is not zero 
in RS) ; therefore, the width-length ratio should 
be kept lower in RS than in ADP transducers. 

Lacking data for detailed verification of the 
theory for the higher modes, we confine our 
attention to the first mode. The resonance fre¬ 


quency of the first mode in RS at room temper¬ 
ature, on the basis of UCDWR data, is: 

RS, first resonance: 

frL = 46.3 - 

= (46.3) (1 - 0.24r2). 

From the coefficient, 0.24, one finds for the 
width Poisson ratio, 

RS: 6, = 0.76. (64) 

Considering the probable error in the coefficient 
0.24, this is regarded as excellent agreement 
with the value of 0.776 calculated® from the 
crystal constants as determined by Mason. 


Piezoelectric Coupling Coefficient 


From equation (38) of Section 3.2.2, the dif¬ 
ference between the antiresonance and reso¬ 
nance frequency, for the various modes, is given 
by 




(65) 


= 1 -f an^rl ISn^T^t, (66) 

“ - ll2 j L-(TTU)-+ J’ 


( 68 ) 


Thus, the variational principle predicts that the 
piezoelectric coupling coefficient will depend 
upon the width-length and thickness-length 
ratios, with coefficients that depend upon the 
modal number. It is readily shown that C^l and 
Cg} are negative, whereas Cl} is positive. This 
means, at least for ADP in which is zero and 
probably also for RS, that F+^ increases with 
increasing and decreases with increasing r,. 
This behavior, for = 1, is qualitatively veri¬ 
fied, for ADP at small and r^, by McSkim- 
If these data follow equations (65) to 
(68) closely, then one could determine the 
values of F-/oc, a and (3. 

Time has not permitted a detailed analysis 
of these data. Furthermore, it is believed that 
the analysis should be done by one intimately 
familiar with the experimental procedures; this 
is because the impedance of a crystal becomes 
very large at antiresonance, and any small 







92 


COMPONENT PARTS 


shunt, arising from stray capacitance or leak¬ 
age resistance, although negligible at resonance 
will cause a serious error at antiresonance. 

We therefore adopt values given by Mason^^ 
and Kinsley°“ for the electromechanical coupling 
coefficient, expressed in terms of Mason’s D 
(our F is Mason’s KD /^%), 

ADP: D = 12.2 X 10^ dynes esu charge (69) 

RS: D = 11.2 X 10^ dynes/esu charge (70) 

These are valid only for the infinitely thin crys¬ 
tal, and, as remarked above, while McSkimin’s 
data agrees qualitatively with the shape correc¬ 
tions predicted by the variational principle, we 
do not yet know if the theory is correct quan¬ 
titatively. 

From equations (69) and (70) together with 
the dielectric constants given in the next sec¬ 
tion, we obtain 

ADP: F = 13.8 X 10^ dynes/esu charge (71) 
RS; F = 8.9 X 10^ dynes/'esu charge (72) 
The dimensionless number. 


characterizes the smallness of the electrome¬ 
chanical coupling. In this formula, y is the 
quantity called the electromechanical coupling 
coefficient by Mason^ and designated by him 
as k, except that qc“, the short-circuit Young’s 
modulus, is here used instead of the insulated 
Young’s modulus, the latter being called Yo by 
Mason. The fractional difference between y and 
k is of order y-‘, the above quantity appears 
to emerge more directly from the analysis 
and, in any event, a careful study shows that 
unless one takes proper account of width cor¬ 
rections for all the parameters, which has not 
yet been done in numerical detail, fractional 
differences of this order are of no importance. 

The numerical value of y is about 0.3 for both 
ADP and RS. This is the basic reason for the 
narrow band width of crystal transducers, as 
may be seen by noticing that the product of the 


mechanical and electrical Q’s for ADP and 
RS is 

and that the optimum band width is determined 
by the two Q’s being approximately equal, so 
that the optimum effective Q is of order (14)’'^ 
or about 4. This figure refers to a single crystal 
but in an actual transducer this product, and 
therefore the optimum effective Q will be larger 
because shunt capacitance increases Q^j above 
its single crystal value (also, the effective Q 
may be larger if Q,^ is low because of poor radi¬ 
ation loading). This does not affect the product 
above, but the two Q’s are then not approxi¬ 
mately equal, as they happen to be for full water 
loading. 

Dielectric Constant 

From Section 3.2.2, where the low-frequency 
limit of the equivalent circuit is discussed, one 
sees that at a frequency well below resonance, 
the circuit simplifies to two parallel condensers, 
the blocked electric condenser and the equiva¬ 
lent (low-frequency) mechanical condenser. 
This equivalent low-frequency mechanical con¬ 
denser should not be confused with the approxi¬ 
mate condenser used with a similar inductance 
to represent the circuit near resonance. 

The ratio of the low-frequency mechanical 
condenser, in electrical units, to the blocked 
electric condenser, is y“ [see equation (32) 
of Section 3.2.2]. Therefore, to measure the 
blocked capacitance one needs only, in principle, 
to measure the total electric admittance and 
apply a correction factor, 

(1 + 7^)-^ ^1-77 (75) 

to eliminate the mechanical condenser. 

However, one is measuring a very low capac¬ 
itance, and strays therefore become very impor¬ 
tant, so that elaborate precautions^ must be 
taken to get reliable results. Measurements at 
this laboratory check those done at other lab¬ 
oratories but have considerable dispersion 
(about ±10 per cent) and are therefore re¬ 
garded as inferior to those obtained elsewhere. 
We therefore adopt the values given by Mason 






SINGLE CRYSTAL PLATES 


93 


for the longitudinally blocked dielectric capaci¬ 
tance of ADP^ and RS^'’: 


ADP: 

K = 14.2, 

(76) 

RS: 

K = 10.0. 

(77) 


The variational principle predicts that this 
quantity has no appreciable finite width correc¬ 
tion (see Section 2.5.3). 

Summary of the Perturbed 
Equivalent Circuit 

For convenient reference, the constants and 
perturbed parameters are summarized in Table 

2 . 


Electrodes 

In order to drive all parts of the crystal with 
an electric field which is approximately uni¬ 
form, the large faces of the crystal plate are 
covered with conducting sheets, called the 
electrodes. 

Various types of electrodes have been used 
by different laboratories, A description of these, 
together with the techniques involved, is given 
in Chapter 8. Here we are concerned with the 
influence of the electrode upon the measurement 
of crystal constants and the performance of 
completed transducers. 

The electrode believed to most nearly approx¬ 
imate a perfect conducting sheet exerting neg- 


Table 2. Constants for infinitely thin rectangular plates of 45° Z-cut ADP and 45° Y-cut RS. 


Symbol 

Constant 

Units 

ADP 


RS 


Q 

Density 

g 'cm3 

1.80 


1.78 



Y 

Short-circuit Young’s modulus 

dynes/cm2 

1.95 

X 1011 

0.98 

X 

1011 

c 

Phase velocity 

cm/sec 

3.29 

X 105 

2.35 

X 

105 

Zc = QC Characteristic impedance 

g/cm2 sec 

5.92 

X 105 

4.17 

X 

105 

F 

Blocked stress per unit field or short 

dynes/esu charge 

13.8 

X 104 

8.9 

X 

104 


circuit charge density per unit 
strain 







K 

Longitudinally clamped dielectric 

dimensionless 

14.2 


10.0 




constant 







tw 

Width Poisson ratio 

dimensionless 

0.7 


.78 




Thickness Poisson ratio 

dimensionless 

0.2 to 0.3(?) 

(?) 



e. 

Shear Poisson ratio 

dimensionless 

0 


(?) 



Y 

Electromechanical coupling coeffi¬ 

dimensionless 

0.3 


0.3 




cient 








Some of these values are undoubtedly in error 
by a few per cent, but until transducer design 
is greatly refined, they are accurate enough for 
practical purposes. 

The perturbed parameters are given in Sec¬ 
tion 2.5,3. To use these, they should be evalu¬ 
ated at the resonance near which the transducer 
is used, which amounts to setting 0 — kL equal 
to njt/2 or nn for a crystal which is approxi¬ 
mately blocked or approximately free-free, with 
w 1, 3, 5, etc. 

The finite width correction to the resonance 
frequency has been worked out in detail, but 
that for the piezoelectric constant F+ has not. 
The values of the tangential and normal im¬ 
pedances to be used in these formulas are not 
at present known. 


ligible influence on the motion of the crystal is 
a very thin layer of gold applied by an evapora¬ 
tion technique developed at Bell Telephone Lab¬ 
oratories (see Chapter 8). This electrode is 
therefore used for the evaluation of crystal 
constants. 

The elastic and dissipative effects of other 
electrodes are greater. However, if the tech¬ 
nique of attaching them is good, they all appear 
to have negligible effect upon the performance 
of transducers, the slight additional effects 
being overwhelmed by dissipation of other ori¬ 
gin and by radiation resistance. This is readily 
verified by measuring the shift in frequency 
and the mechanical Q of a single crystal 
equipped with various electrodes. These quan¬ 
tities differ from those obtained with the evap- 








94 


COMPONENT PARTS 


orated gold electrodes by amounts which are 
significant with respect to determining crystal 
constants, but entirely negligible with respect 
to the performance of an actual transducer. An 
exception to the above may occur in the case of 
electrodes which are not cemented to the crystal. 
In this case, it is believed that castor oil be¬ 
tween the crystal face and the electrode may 
introduce a tangential resistance of practical 
importance. 

Three electrical effects require consideration; 
(1) the loss arising from the resistance of the 
electrode, (2) the stray capacitances from the 
electrodes to each other and to the backing plate 
and to the case, and (3) the effect of the con¬ 
densers formed by the cement layers. 

The resistance of the electrodes seems to be 


the impedance seen looking in at these ter¬ 
minals, thereby increasing the effect of the 
shunts arising from the stray capacitances to 
the backing plate and to the case. 

To estimate the importance of these cement- 
layer condensers, suppose the cement layers are 
identical and have thickness t' and dielectric 
constant K'. The ratio of the voltage drop in 
the cement layers to that in the crystal, given 
approximately by the inverse ratio of the ca¬ 
pacitances, is 

(I) (f)- 

Using rough figures for K' and t' of 2 and 1 mil, 
respectively, which are extreme examples, this 
is about 0.05 for a crystal i/i in. thick, and 0.25 



negligible since the series equivalent resistance 
of a single crystal working into water is several 
hundred kilohms. 

All the electrodes have stray capacitance to 
others and to other metal objects in the case. 
This is a purely electrostatic phenomenon, de¬ 
pending on clearances and the dielectric mate¬ 
rials which find themselves in stray fields, and 
hence will vary with the type of electrode used 
only in so far as the various techniques lead to 
different geometry (see Section 4.5 and Chap¬ 
ter 8). 

The capacitances corresponding to the cement 
layers enter the equivalent circuit as shown in 
Figure 12. These condensers reduce the field in 
the crystal, and reduce the band width in two 
distinct ways: (1) they increase the electric Q 
as seen looking in at the electrode terminals 
(+ and — in Figure 12), and (2) they increase 


for a crystal 0.05 in. thick. If the dielectric con¬ 
stant K' is 5 instead of 2, the effect is only 40 
per cent as great; likewise, a cement layer of 
0.5 mil will halve the effect again. This indicates 
that, judging solely from this consideration, it 
is advantageous to use a cement of high dielec¬ 
tric constant and to make t' as small as possible 
by using high pressure and a cement which 
flows easily. This is done in most cases. 

We conclude that in most applications, the 
cement layer is not very important, but that its 
importance is increased if the crystal is X-cut 
RS (large K), if the cement has low dielectric 
constant, or if the crystals are thin. The use of 
thin crystals appears to be important for high 
power, and here also any dissipation in the 
cement layer, while not important from an effi¬ 
ciency viewpoint, may be important in the 
breakdown mechanism. In this case, the desira- 



















SUBASSEMBLIES 


95 


bility of evaporated gold electrodes should be 
considered carefully. 


Weakened Crystals 

Some fragmentary experiments have been 
done by UCDWR on the weakening of crystals 
to obtain low resonance frequency without 
going to undue length. 

Various methods of weakening were used, 
including boring holes normal to the electrode 
faces and sawing slots in from both secondary 
radiating faces in planes perpendicular to the 
longitudinal axis. 

In this manner, a 1/4x1x11/2 in. 45° Y-cut RS 
plate has been made to have a resonance fre¬ 
quency as low as 2 kc, to be compared with its 
unweakened value of 27.5 kc. It is, of course, 
possible to obtain any frequency between these 
values. 

Although the resultant crystals were not as 
fragile as one might think, they nevertheless 
could not be trusted without thorough field test¬ 
ing and the pressure of other work during 
World War II did not permit this. There is still 
no entirely satisfactory design for crystal trans¬ 
ducers to operate below about 10 kc, and hence 
it might pay to investigate the possibility of 
building transducers of satisfactory ruggedness 
with weakened crystals. 


33 SUBASSEMBLIES 

Cement Joints 

The satisfactory attachment of crystals to 
other surfaces is one of the most critical steps 
in the construction of transducers. The tech¬ 
niques are discussed in Chapter 8, and only the 
desirable properties of joints and the theory of 
measuring their impedance are discussed here. 

Some of the most direct and useful tests are 
purely empirical. For example, a cement joint 
may impose a tangential constraint such that 
differential expansion will cause the crystals to 
crack at low temperatures, and only an em¬ 
pirical test can settle this question. Again, a 
joint may deteriorate if forced to transmit high 


acoustic intensity, and the only practical way 
of detecting such behavior is to measure its 
impedance preferably while driving (an experi¬ 
mental technique not yet developed) or a very 
short time before and after such hard driving. 

The ideal joint is one which has zero tan¬ 
gential and infinite normal impedance. Failing 
this, it is believed that the most important single 
item is that the normal impedance be uniform 
from joint to joint, since nonuniform joints on 
a backing plate or bar are believed to excite 
flexural modes which distort patterns and cause 
serious losses. Naturally, one would like the 
resistive parts of both normal and tangential 
impedance to be small enough to cause negli¬ 
gible losses; however, a tangential reactance, 
if not too large or variable, is not believed to 
be serious, since it would merely cause a slight 
alteration in the resonance frequency by stiff¬ 
ness coupling through the Poisson motion. 

The most sensitive means for finding the im¬ 
pedance of a joint consists in cementing a crys¬ 
tal to an approximately quarter-wave backing 
rod and observing the frequency and the me¬ 
chanical Q, being careful to exclude all extrane¬ 
ous dissipation. From equation (16) of Section 
3.2.2, evaluating the numerator of Z\ at the 
unperturbed resonance, one has for the imped¬ 
ance of the mechanical arm 


— iZ+ cot 0+ + 



(79) 


in which Z\ and 0“^ are real since it has been 
assumed that precautions have been taken to 
make internal dissipation negligible. One now 
regards Z| as consisting of a reactance and 
a resistance R]_ in parallel, 


J_ = 1 _ _L 

Z| Ry Ah • 


(80) 


The deviation of the observed resonance fre¬ 
quency from that calculated from the free-free 
resonance of the crystal, taking account of the 
different finite width corrections, gives 




lll£ 


(81) 


and the resistance is determined from the me¬ 
chanical Q, or from the absolute admittance 
(see Chapter 9). The determination of Ai ad- 





96 


COMPONENT PARTS 


mittedly rests upon the small difference between 
an observed and a calculated frequency. How¬ 
ever, if the difference is so small as to cause a 
large error, then Xi is, for all practical pur¬ 
poses, infinite. 

Many tests of this type were made in this 
laboratory during the progressive baking of 
cement joints, and the observed frequency ap¬ 
proached a limiting value, as the baking time 
became long, corresponding to a very high im¬ 
pedance. Simultaneously, increased, indicat¬ 
ing an increase in the parallel resistance. Sim¬ 
ilar tests were also made to determine the effect 
of driving the joint very hard (to cavitation for 
various times), and some types of joints showed 
progressive deteriorations. This procedure was 
very helpful in finding a technique which yielded 
high Q joints that stood up. 


Crystal Blocks 

Single crystals can be cemented together in a 
variety of ways, all of which will be referred 
to as crystal blocks. 


cemented into blocks, the polarity being care¬ 
fully matched. The results are shown in Table 3. 

These crystals were taken at random from 
routine production, and were equipped with 
0.001-in. German-silver electrode foils. The 
small discrepancies in the resonance frequen¬ 
cies of the single crystals are undoubtedly 
caused by slight nonuniformities in the elec¬ 
trodes. The frequencies of the blocks were cal¬ 
culated as if they were single crystals, and it 
will be seen that the discrepancies are only a 
little larger than those for the single crystals. 
The mechanical Q’s for the blocks were of the 
order of 500 as compared with 1,000 for the 
single crystals. The crystal blocks were driven 
to cavitation for 2 hr and measured again, with 
the result that neither the resonance frequen¬ 
cies nor the Q’s changed within experimental 
error. 

These results indicate that crystal blocks with 
good shape factors may be regarded as single 
crystals for all practical purposes, provided the 
cementing technique is good. It should be em¬ 
phasized that the above blocks have width- 
length ratios appreciably less than 1. Some 


Table 3. Effect of cementing crystals together on secondary radiating faces, and on their electrode faces. 


Dimensions 

(inches) 

Single Crystals 
Resonance frequency (kc) 
Calculated Observed 

Number of 
Crystals 

Dimensions 
of block 
(inches) 

Crystal Blocks 

Resonance frequency (kc) 
Calculated Observed 



A. Secondary radiating faces. 



14X1/2X1" 

61.5 

61.6 ± .05 

3 

14x11/2X1" 

39.1 

39.2 ± 0.1 

14x1/^x114" 

50.0 

50.1 zt .05 

2 

14x1x114" 

44.8 

44.5 ± 0.1 



B. Electrode faces. 



14x14x1" 

61.5 

61.6 ± .05 

2 

1/2 X 1 / 2 X 1 " 

... 

61.25 ± 0.5 


It is sometimes convenient to cement crystals 
together on their primary radiating faces, to 
obtain crystals larger than those which can be 
economically grown (45° plates are cut as 
shown in Figure 1 of Chapter 1, and the cost 
of crystal plates therefore increases with the 
length). It has been demonstrated that such 
fabricated crystals are in all practical respects 
the equivalent of a single crystal of the same 
dimension, provided the cementing technique is 
good and the proper polarity is observed. 

To determine the effect of cementing on the 
secondary radiating faces and electrode faces, 
a number of 45° Z-cut ADP crystal plates were 


fragmentary results indicate that the motion 
of blocks (as well as single crystals) becomes 
quite complicated as the width-length ratio 
approaches 1. This would be expected theoret¬ 
ically, and is confirmed by the silicon carbide 
dust pictures. Figure 12 of Section 2.5.3. 

The result for cementing on the electrode 
faces (Table 3) gives 0.2 as a very rough esti¬ 
mate of the thickness Poisson ratio. 

On the basis of the foregoing admittedly 
scant data, it has been the policy of this labora¬ 
tory to use blocks whenever necessary to get 
needed resonance frequencies from available 
crystals or whenever desirable to simplify con- 









SUBASSEMBLIES 


97 


struction. The satisfactory performance of 
many transducers using blocks cemented on 
their electrode faces, and of a few using blocks 
cemented on other faces, is partial and indirect 
confirmation of the above results. 


" " " Benioff Blocks 

An extremely rugged transducer, designed by 
Dr. Benioff for Submarine Signal Company, 
employs a structure known as a Benioff block. 
This consists essentially of a crystal working 
into a backing and a fronting rod, the two rods 
being held together by a quite rigid tie rod 
which opposes the longitudinal vibration of the 
crystal. 

The only feature of this structure that will be 
discussed here is its band width. The equivalent 
circuit for the block can be approximately for¬ 
mulated from equivalent circuits for the crystal, 
the tie rod, and the backing and fronting rods. 
In the absence of dissipation, Foster’s theorem 
tells us that the slope of the reactance curve 
for the mechanical arm will be increased at all 
points, including resonance. Now inserting the 
radiation resistance, we conclude that the me¬ 
chanical Q will be greater than for a crystal 
without constraints imposed by the tie rod. This 
conclusion is borne out by experiment, the Q 
obtained from the series resistance curve in 
water being of order 200 as compared with 
3 to 6 for most transducers. 

A careful study of the Benioff block has not 
been made at this laboratory. The foregoing 
brief discussion gives a qualitative theoretical 
explanation of its observed sharpness and this 
indicates that a structure of this type will not 
be satisfactory for wide-band operation; how¬ 
ever, it is possible that it may have practical 
applications for single-frequency operation. 


Unit-Construction 

A unique method of mounting crystals has 
been developed at the Naval Research Labora¬ 
tory [NRL]. This consists in mounting a small 
number of crystals on a single cylindrical back¬ 
ing rod with a rectangular plate across the top. 


and then isolating the rod by a cylindrical cup 
attached to the rod near its node by a rubber 
bond; these cups are attached to the supporting 
plate. The basic principle is indicated in Fig¬ 
ure 13. 

Performance data on unit-construction trans¬ 
ducers are not available at the time of writing. 


RUBBER BOND CRYSTALS 




I!' 


't. 

! 

//' 



A 


B 


Figure 13. Unit-construction (NRL). 


However, one would think that this typ'e of 
construction has much to recommend it. 

From the design viewpoint, it would appear 
to eliminate all troubles arising from flexural 
modes in a backing plate (see Sections 3.4 and 
3.6). A secondary design advantage is that it 
replaces any loss arising from radiation from 
the backing rod by a loss in the rubber bond, 
which is probably much smaller since this bond 
is so near the node. However, the crystals oper¬ 
ate in castor oil, just as in many other trans¬ 
ducers, and the motion of their radiating faces 
is greater than the free end of the backing rod 
in the inverse ratio of the impedances, about 
6 for steel and 2.5 for aluminum; since the loss 
is proportional to the squares of these ampli¬ 
tudes, we see that the crystal loss is by far the 
more important, and one does not gain much in 
reducing the smaller loss by using the cups. 

From the production viewpoint, one can fore¬ 
see a great advantage. Production testing is 
always a difficult problem, and it is not simple 
to maintain standards. With this type of con¬ 
struction, much of the critical work would be 
done on small objects susceptible to rapid and 
accurate testing. This is an advantage scarcely 
to be overestimated. It is therefore believed 
that the possibilities of the unit-construction 
transducer should be very thoroughly studied 
and, unless unforeseen difficulties arise, that 
many types of transducers should use this kind 
of construction. 
























98 


COMPONENT PARTS 


3 ^ BACKING RODS, BARS, AND PLATES 

As the entire surface of an excited crystal is 
in motion, the problem of supporting it without 
interfering with the motion is one of impor¬ 
tance. A common practice is one of cementing 
one end of the crystal to a metallic rod or bar, 
or an array of them to a flat plate. The condition 
for resonance of a system of a block of crystal 
cemented to a block of metal, is, as given in 
Section 7.6 

-2c tan {kL)c = Zb tan {kL)b, 

where c and h indicate crystal and backing 
plate, respectively, and 2 ^ is about 5 X 10" and 

for steel is about 39 X 10". Thus for a given 
frequency, the crystal and backing material 
lengths can be adjusted to any suitable value. 
This construction allows a much shorter crystal 
to be used for a given frequency, which is an 
advantage in saving crystal material. If, for 
example, using Y-cut RS, it is desired to design 
a transducer to be resonant at 40 kc and the 
width and thickness of the units are thin enough 
to have no effect on the longitudinal velocity, 
and each crystal is backed by a bar whose di¬ 
mension ratios have the same property, the 
following lengths of crystals and backing bars 
can be used, all of which will resonate at 40 kc. 


pling between the crystals can be very trouble¬ 
some, in that it produces uneven end velocities 
and phases, which in turn make for bad direc¬ 
tivity patterns, poor efficiencies, etc. To com¬ 
pletely eliminate this coupling, all crystals 
should be completely isolated from each other, 
and this has been most nearly done in the case 
of Cycle-Welded transducers (Chapter 8). How¬ 
ever, many transducers have to be built on 
backing plates, etc., so that the coupling prob¬ 
lem must be dealt with. For individual backing 
of each crystal, the backing bars must be me¬ 
chanically connected in some way in order to 
hold the array together. One method is to bolt 
them all on a thin plate. An example of this 
construction is the Navy-type QBF transducer, 
(see Section 3.5). The coupling in this case is 
reduced to certain frequencies out of the oper¬ 
ating band of the transducer. The theory of this 
isolation is given later in this section. 

The most troublesome coupling is in the types 
where all the crystals are cemented upon the 
same steel plate or bar. Flexural modes of the 
plate or bar are very many indeed, they usually 
lie in the operating-frequency band, and their 
direction of vibration is that of the crystals. 

The natural frequencies of flexure of a rec¬ 
tangular bar of width thickness a and length I, 
whether bounded or unbounded, are given by 


Bar length (inches) 0.25 0.5 1.0 1.5 2.0 

Crystal length (inches) 0.79 1.10 1.30 1.38 1.42 

The above formula was developed on the as¬ 
sumption that only the fundamental longitu¬ 
dinal vibrational mode is present in both the 
crystal and the backing bar, and this assump¬ 
tion necessitates both elements to be long and 
thin. When the length approaches the thickness 
in a bar, the frequency of the first mode is 
lowered, and when the length is small compared 
to the cross section, this frequency is lowered 
to less than one-half that of the long bar of the 
same cross section. A graph of the reduction of 
frequencies with thickness appears in Chap¬ 
ter 4. Another way of looking at the problem 
is to consider the velocity of sound in the bar 
to vary with thickness. The formula 2 ^ tan ikl) ^ 
is still good if for k = 27iv/c the value of c is 
used that is present in the thick bar. 

In an array of crystals, the mechanical cou¬ 


wk / E o 

I3n = 1.505, 2.50; /3„ = + |. 

For steel this reduces to 


(82) 


y„ = 2.44 X 10’I 

The thickness a is determined by the operating 
frequency of the transducer, the crystal mate¬ 
rial, etc., and is usually of the order of 1/2 in. to 
11/2 in. It is easily seen that if the first mode is 
to be around 10’ c the length must be no greater 
than about three times the thickness a. This 
condition restricts the bars to be smaller than 
is sometimes desirable. Above the first or grav¬ 
est mode, the higher-modes appear in abun¬ 
dance, and in addition the vibration patterns 
are quite complex due to the superposition of 
torsional and tangential modes. The modes of a 
steel bar %q in. wide, % in. thick, and 8 in. long 





BACKING RODS, BARS, AND PLATES 


99 


were studied with the probe technique discussed 
in Chapter 9, and are reproduced in Figure 14. 
The calculated flexural frequencies occur at 
2.12, 5.80, 11, 20.2, 30.2, 42.0, etc. These fre¬ 
quencies are indicated on the figure. The other 
modes observed and interspersed among these 
are torsional, as the phases of their motions 
shifts 180° as the probe moves across the bar 
surface perpendicularly to the length. 

The calculations of the flexural modes in a 
backing plate involves a two-dimensional wave 
equation of fourth order similar to that of the 
bar, the solution of which is straightforward, 
but in applying the boundary conditions of no 
restraint, considerable difficulties arise in the 
case of the square plate because of the compli¬ 
cated stresses set up close to the free edges.® 
The case of a circular boundary has been solved' 
giving the allowed frequencies as 

fmn = ^2 j/ 3p(l _ s2) (83) 

where 

/3oi = 1.015 /3 o2 = 2.007 t = half-thickness, 
a = radius, 

i8n = 1.468 iSi2 = 2.483 E = Young’s modulus, 
p = density, 

/32 i = 1.879 = 2.992 s = Poisson’s constant 

(about 0.33). 

The gravest mode thus for steel occurs at 
/oi = 4.86 ^ X lOK 

The lowest mode for a steel plate 1 in. thick 
and 6 in. radius occurs thus around 5 kc. The 
higher modes are spaced at frequency intervals 
of only a few kilocycles, and exhibit modal cir¬ 
cles and diameters of many patterns. 

A collection of modal patterns exhibited by a 
square plate 1/2 in. thick by 4 in. by 4% in. 
observed with the probe technique is given in 
Figure 15. In each case the drive is concen¬ 
trated at one corner as indicated. These pat¬ 
terns are complicated with superpositions of 
several modes at once and are practically im¬ 
possible to calculate. It is significant, however, 
that the frequency range covers quite thor¬ 
oughly the entire range from 3 to 100 kc. How¬ 
ever, all of them are not necessarily excited in 
a transducer, because a particular mode is ex¬ 
cited only by a particular exciting force. With 


a completely uniform drive at a mode fre¬ 
quency, the mode will not be excited, but if the 
drive is nonuniform (and most of them are 
nonuniform) some evidence of the flexural 
mode of the backing plate appears, and may 
range in importance from negligible to the 
controlling factor. Two such effects have been 
noticed, one in probe studies, the other in di¬ 
rectivity patterns. Figure 16 shows some probe 
studies of a motor-crystal face close to the 



90 


Figure 14. Flexural modes of steel bar, 8 x%x%g 
in. 

frequency of a backing-plate mode. The surface 
distributions charted there are those of phase, 
showing that in each case the uniform phase 
distribution is disturbed by the backing plate at 
its mode frequency. In directivity-pattern 
studies there are many examples of erratic be¬ 
havior in backing-plate transducers at the 
higher frequencies, while other types (see Sec¬ 
tion 4.3) usually are not as erratic. At lower 
frequencies also, whimsical directivities are 



















100 


COMPONENT PARTS 









1 t 1 1 

lilt 1 

1 1 

1 1 1 

1 1 1 

1 1 1 

t 1 1 

1 1 1 

1 1 

till 1 

’►♦ + + + 

+ 4^ + ♦ 

♦ + + + 

+ 

+ + ♦ ♦ 

+ 

+ 

+ + + + 

+ + + + 

+ + + •♦■ + 

+ *♦• + + + 

+ + 

+ + + 

♦ ♦ 

+ + 

+ + 

+ 

•► + + + + 

+ + + + ••■ 

1 1 1 

1 1 

1 1 1 

1 1 

1 1 I 

1 1 

1 * 1 

1 1 

1 1 1 

- - DRIVE 

- JJOKC 


Figure 15. Flexural modes of square backing plate 4x4%xV^ in 









































































































































































































































BACKING RODS, BARS, AND PLATES 


101 


-W - Vt)- 





BACKING PLATE SURFACE 




36P0 KC 37.55 KC 4C100 KC 

CRYSTAL MOTOR FACE 



3U7S KC 33.42 KC 33^ KC 

CRYSTAL MOTOR FACE 




BACKING PLATE SURFACE 




ISOO MC 21.21 KC 21^75 KC 

CRYSTAL MOTOR FACE 



6CL24KC 68.56 KC 7000 KC 

CRYSTAL MOTOR FACE 


Figure 16. Evidence of backing-plate modes on GD28 crystal motor. Phase legend: + indicates 180° 
phase, 0 indicates 90° phase, — indicates 0° phase. 


noticed which in some instances have been 
traced to irregular surface velocities and phases 
caused by backing-plate flexural modes. 

If deep slots are cut in the back of the plate, 
flexural modes may be suppressed at least over 
a frequency band depending on the dimensions 
of the cuts and their spacings. The theory of 
this suppression is as follows. 

If a rectangular bar is slotted transversely 
so that the remaining web is a tenth of the bar 
thickness, and these slots cut out sections of the 
bar short compared to the wavelength of flex¬ 
ural modes within the bar at a particular fre¬ 
quency, the web can, with good approximation, 
be thought of as a massless compliance and the 
remaining block as compliantless mass as shown 
in Figure A. 

Under these assumptions the bar becomes an 


acoustic low-pass filter for flexural waves whose 
electric equivalent is given in the figure. The 
circuit constants can be calculated by the 
method illustrated in Figure B. Each mass is 
joined by a thin web which under transverse 
vibrations acts as a beam undergoing a strain, 
the type of which is illustrated in Figure C. The 



-6- 


-e- 


-6- 













M 


M 


M 


M 


M 


^-^00 

' y T _I_I_J 


Figure A. 













































































































































































































102 


COMPONENT PARTS 


vertical compliance of such a beam is calculated 
to be 

i2Er 

where L is the length of the beam, E is Young’s 
modulus, and I is defined as 

r + to/2 

I = w \ m, 

integrated about the central unstrained hori¬ 
zontal plane in the bar. The cutoff frequency of 
this filter occurs at 


WMCm' 

Putting in the value of C,„, in which the value 
of I calculated for a rectangular cross section 
is substituted, the cutoff frequency becomes 





(84) 


Substituting in the values of E" rr: 2 X 10^-, 
Q = 7.7, for steel, the cutoff frequency becomes 


fc 64.4 |/ ^3, 


where is in kilocycles, T, d are in inches. 


J 

1 

M 

1 

1 

t 

i 



«*-d 



Figure B. 



Specific Examples 

QBF-Type Backing System. The backing sys¬ 
tem of the QBE transducer consists of a in. 
steel plate to which steel bars IVs in. square and 
2 in. long are bolted and soldered. They are 
spaced % in. from each other on all sides. A 


single line of these elements is illustrated in 
Figure D. The cutoff frequency of this line cal¬ 
culated from the formula developed in the pre- 



Figure D. 

ceding section is 22 kc. This is just 2 kc below 
the frequency at which the unit is designed to 
operate. 

Slotted Square Bar Backing System. This bar 
(Figure E) showed no eigen modes between a 



band of 30 to about 58 kc. At this upper fre¬ 
quency, the wavelengths are approaching the 
size of the blocks, so that complicated modes 
begin to appear. As the frequency is yet raised, 
the modes get very complicated and very numer¬ 
ous. A spectrum of frequencies of the modes is 
shown in Figure 17. 

The spectrum of another bar similar in every 
respect except that the connecting webs are 
twice as thick (%o hi.) is shown in Figure 17. 
The region of suppressed modes is much nar¬ 
rower in this case, showing that the web is less 
of a massless compliance than is the thinner 
web, and emphasizing the necessity of keeping 
the webs thin. A square plate was slotted in sec¬ 
tions of the same dimensions as the first bar 
above, whose eigen modes appear in Figure 17 
as the lower spectra. It is almost impossible to 
compute the frequencies of these modes, as the 
calculation involves the problem of the un¬ 
bounded square plate. However, experiment 
shows a suppressed region for the plate as for 
the bars. 

In conclusion it can be said that the best way 






































































MULTIPLE-LAYER BACKING PLATES 


103 


to get free of flexural modes in the motor is to 
isolate all crystals with their individual back¬ 
ing terminations from each other. If this is not 
feasible, flexural modes can be suppressed only 
within a moderate frequency range. 

3- MULTIPLE-LAYER BACKING PLATES 

Situations occasionally arise in which a back¬ 
ing plate is required, but the thickness or 


structure may be described adequately by a 
transmission-line treatment (which amounts to 
the assumption of pure plane-wave motion). 
The internal losses in materials likely to be used 
in backing plates are so low that the assump¬ 
tion of a dissipationless line is justified; this 
simplifies the theory somewhat as hyperbolic 
functions revert to ordinary trigonometric 
functions. 

A single layer of some lossless material may 
be represented by the equivalent circuit shown 





5 10 15 20 30 40 50 60 70 80 

FREQUENCY SCALE IN KC 

Figure 17. Mode spectra of slotted backing bars and plate. 


weight of an ordinary plate exceeds allowable 
limits. Provided restricted band width is accept¬ 
able, such a situation may be aided by the use 
of a multiple-layer backing plate. 

Consider first the two-layer system shown in 
Figure 18. The crystal array is cemented to a 
plate of thickness Li, characteristic impedance 
Zi, in which the velocity of sound is Ui. This 
plate is perfectly attached to a second plate of 
different material whose thickness is Lo, char¬ 
acteristic impedance Zo, and velocity of sound 
Fo. This second plate is then terminated by air 
or other substance whose characteristic imped¬ 
ance may be considered zero. 

If the motion of the crystals is simple, uni¬ 
form, and in phase, and if flexural resonances, 
etc., in the plates are successfully avoided the 


in Figure 19, where Zq is the characteristic im¬ 
pedance of the material, and 

G - y, 

where L is the thickness of the layer and V is 
the velocity of sound. 

It is easily shown that if an arbitrary com¬ 
plex impedance Z^ is imposed on one end of this 
line the impedance Zj seen looking into the 
other end of the line is 

7 _ 7 Zt cos 6 + yZo sin d 
" ° Zo cos d + jZr sin 6’ 

Note in passing that if the thickness and fre- 










































































































104 


COMPONENT PARTS 


quency are such that the plate is exactly a quar¬ 
ter wave thick then 



and ZiZt = Z%. 

In particular, if a lossless quarter-wave line is 
terminated by zero impedance the input imped¬ 
ance at the other end of the line (other face of 
the plate) is infinite. This is the basis on which 


Returning to the problem presented in Figure 
18 and using the equation above, one may calcu¬ 
late the impedance seen looking into plate No. 
2 from the interface between the plates. The im¬ 
pedance so calculated then becomes the termi¬ 
nation impedance imposed on plate No. 1, and 
the impedance seen by the crystals looking 
backward into the plates is readily obtained. 
The result is: 

7 _ -7 ^2 tan 02 + ^1 tan 

- ./A loo; 



Figure 18. Typical two-layer backing plate. 

ordinary backing plates are chosen. Note also 
that if the plate is an eighth wave thick and 
terminated by air then 

Zj = -\-jZo. 

Since the characteristic impedance (Zq) of steel 
is much greater than that of crystals, an eighth- 
wave plate provides a sufficiently high imped¬ 
ance to nearly clamp the crystals. 

Note finally that if the layer is a half wave 
thick (or any integer multiple of a half wave) 
then 

Zj = Zj.. 

This indicates that any integer number of 
half-wave sections may be “lifted out” of a loss¬ 
less line without changing the impedances at 
the ends. 


The question now arises: under what condi¬ 
tions is infinite? Obviously solutions will be 
obtained when 

Zi = Zi tan di tan 02 (87) 

(with reservations concerning the behavior at 
the poles which, it will be seen, are of no prac¬ 
tical importance). 

For any given co there are an infinite number 
of Li, Lo combinations which satisfy this condi¬ 
tion. Obviously the quantity Li -f- Lo (where Li 
and Lo satisfy the condition) will be some func¬ 
tion of Li, and the problem is now to select the 
optimum Li, Lo combination. 

The problem is solved here on the assumption 


+ i Z^+an ^ + i Z^ton ^ 



Z^ = CHARACTERISTIC IMPEDANCE 


Figure 19. Equivalent circuit of a single layer 
of lossless material. 

that the optimum backing plate will be the 
thinnest backing plate obtainable with any par¬ 
ticular pair of materials at any specified fre¬ 
quency. One might instead require that the 
total mass be minimized, and different conclu¬ 
sions would be reached by analogous methods. 

It is desired to minimize Li -f- Lo, subject to 
equation (87). The method of Lagrange multi¬ 
pliers is used. The theory of this method is 



























MULTIPLE-LAYER BACKING PLATES 


105 


available in standard works on theoretical 
physics.*^ We form the function 


F = Li Lo + A { tan di tan do 




where A is an undetermined multiplier. 
Partially differentiating: 


= 1 + A ^ tan 6-2 sec’ 0i, 

dF 1 , A 4. n 0/1 

= 1 + A -YT tan di sec^ do. 
oLi 2 V 2 

Setting these equal to zero, transposing U’s and 
dividing one equation by the other: 


Vi _ tan 02 sec- di 
Vo ~ tan di sec’ 02* 

It is quite convenient that A cancels out, obvi¬ 
ating the usual necessity of evaluating the mul¬ 
tiplier. 


Substituting, 

S€ 

and 

we obtain 


tan 02 = 


Li = — tan-i 


1 + tan- a 


z. 

Zo 

tan di 

z. 

1 Z, V2\ 

Zo 

(Z2 Vj 


ZiUo 

L Z2U1 


- 1 


Lo = — tan-i 


_ 1 V 
Z 2 1 Z 2 U 1 W 


L Zo 


Z? 


( 88 ) 


(89) 


These expressions serve two useful purposes. 
Firstly they give a criterion for the existence of 
the minimum, and secondly they give numerical 
values of Li and Lo to achieve the minimum. 

To use the functions as criteria, one selects 
two materials of interest such as lead and steel. 
The values of characteristic impedance and ve¬ 
locity of sound are inserted in the expressions 
under the radicals. If the resulting quantity is 
negative no minimum in (Li + L 2 ) exists, and 
the minimum thickness is achieved by 100 per 
cent of whichever material has the lower V. If 
the quantity is positive a minimum does exist, 
and minimum total thickness is achieved by 
using plates whose thicknesses are Li and Lo. 

The criterion concerns combinations of two 


materials, independent of frequency, and a list 
is easily composed telling which pairs of ma¬ 
terials yield a minimum. In applying the cri¬ 
terion it is immaterial which is considered plate 
No. 1 and which plate No. 2. Some results of 
the criterion are: 

Steel; lead: no minimum; use 100% lead. 
Steel; castor oil: minimum exists. 

Steel; aluminum: minimum exists. 

Steel; copper: no minimum; use 100% cop¬ 
per. 

Steel; tin: minimum exists. 

Steel; glass: minimum exists. 

Steel; brass: no minimum; use 100% brass, 
Aluminum; castor oil: minimum exists. 
Aluminum; lead: no minimum; use 100% 
lead. 

Aluminum; glass: minimum exists. 

Lead; castor oil: minimum exists. 

Lead; copper: no minimum; use 100% lead. 
Lead; glass: no minimum; use 100% lead. 
Glass; castor oil: minimum exists. 

It is interesting to note that in more than half 
of these randomly chosen examples the mini¬ 
mum does exist. This should encourage a search 
for other promising pairs, such as glass; sili¬ 
cone fluid. 

When applying the criterion the order of the 
two layers is immaterial. However, it is very 
important that the layers be ordered correctly if 
a minimum does exist. In this case two combi¬ 
nations of Li, L 2 will result from the two orders 
in which the materials may be arranged. One 
of these will be the minimum thickness sought, 
the other will be maximum. This results from 
the fact that Lagrange’s multiplier yields an ex¬ 
treme solution, either maximum or minimum; 
the ambiguity is easily resolved for any pair of 
materials by obtaining both solutions. 

The tan~^ functions are, of course, multiply 
periodic. Examination will show that the peri¬ 
odicity corresponds to the addition of increas¬ 
ing numbers of half wavelengths of material. 
It was shown previously that such a layer con¬ 
tributes nothing, and it is clear that the smallest 
solution is always to be taken. 

It will be made clear by the example below, 
that if a minimum exists, both Li and L 2 will 
be less than a quarter wave thick. This then 












106 


COMPONENT PARTS 


justifies ignoring the poles in the original func¬ 
tion. 

To illustrate the great economy of space pos¬ 
sible by this method, consider the problem of 
producing a clamping backing plate at 24 kc. 
The velocity of sound in steel is usually taken as 
197 X 10^ in. per sec, and a quarter-wave plate 
at 24 kc is 2.05 in. thick. Compare with this a 
two-layer plate using steel and castor oil: 


Steel 

Zo = 39 X 10^ 
^2 = 5 X 10^ 

Oil 

Zi = 1.5 X 10' 
Vi = 1.5 X 10' 
(in cgs units) 


and at 24 kc: CO = 1.51 X 10^. 


The order of materials indicated by the sub¬ 
scripts is chosen to yield a minimum. The values 
of Li and Lo given by the criterion functions 
equations ( 88 ) and (89) are: 

Li (oil) = 0.362 cm = 0.144 in., 

Lo (steel) = 0.334 cm = 0.131 in. 

The total thickness (Li -f Lo) is 0.275 in. as 
compared with 2.05 above: a saving of more 
than a factor of 7. Notice also that although 
(Li -f- Lo) was minimized, rather than mass, 
the weight per unit area is reduced by a factor 
of 13. 

If the steel and oil positions were reversed a 
very thick plate would result. At first sight the 
design problems of providing the oil layer of 
controlled thickness between the crystals and 
the steel seem discouraging. Actually the prob¬ 
lem would be relatively simple if the crystals 
were supported in a jig and the oil layer ob¬ 
tained by shims under the jig. The spacing is 
not quite as critical as it might seem since the 
functions are linear in co. Use of silicone or 
Univis oil should correct any temperature de¬ 
pendences. 

The complexity of the problem increases rap¬ 
idly as more than two layers are used. For one 
thing the Lagrange multiplier does not drop 
out as easily, and the number of simultaneous 
equations increases. Solutions have been ob¬ 
tained for a three-layer system backed by air, 
and the equations for Li and L 3 are the criterion 


functions while Lo acts as a connection func¬ 
tion : 


Li = —tan- 



V 2 . r Z 1 Z 2 - Z 2 Z 3 tan dj tan 63 ~1 
O) L ^ 1-^3 tan ds + Zi tan d, J ^ ’ 



For single-frequency operation there appears 
to be no advantage in more than two layers, and 
aside from the greater engineering complica¬ 
tions the functions seem to be more critically 
dependent on the layer thicknesses. This is par¬ 
ticularly true of Lo which depends on the tan¬ 
gents of 61 and So. It has not been investigated, 
but the third layer undoubtedly offers another 
degree of freedom by which the impedance 
could be adjusted at two arbitrarily chosen 
frequencies. 

It is apparent that this saving is achieved by 
using a pole of a function. Plotting Z^ against 
(0 indicates that the high impedance is obtained 
over a band much narrower than that of a 
single steel plate. Consequently this method is 
restricted to applications requiring a somewhat 
restricted band width. However, the restric¬ 
tions are not severe, and the possibility of quite 
practical transducers invites further examina¬ 
tion of this method. 

A final comment is required concerning the 
physical interpretation of these theoretical re¬ 
sults. Both Li and L^ are much less than 1 / 4 , 
wave in the respective media, so the phase 
change through each plate is small. However, a 
phase change occurs at the interface, and this 
quite large change brings the total phase change 
up to jt/2. This points up sharply a caution not 
always observed: thin layers of material adjoin¬ 
ing layers of very different material may intro¬ 
duce unexpectedly large effects, and a designer 
can ignore such layers only at the risk of con¬ 
siderable error. 











PROBE EXAMINATION OF MOTORS 


107 


PROBE EXAMINATION OF MOTORS 

It is desirable that a transducer present the 
same velocity-amplitude and phase over its en¬ 
tire radiating face to the medium. Many factors 
arise to thwart this ideal, among which are: 
nonuniformity of crystal drives, backing-plate 
flexural modes, cavity modes of transducer 
cases and intercrystal couplings. The effects of 
all these factors working together are noticed 
in the erratic directivities, impedances, and sen¬ 
sitivities of the completed transducers. With 


shows the effect of gluing the array to a back¬ 
ing plate. This array is made up of carefully 
selected crystals, and will be described in detail 
later in this section. Figure 21 shows a contrast 
of measurements of this motor made both in air 
and in oil. Phases as well as amplitudes are in¬ 
cluded in this figure. Figure 22 shows the ve¬ 
locity distribution in air over the face of an¬ 
other motor designed to have two velocity areas 
for directivity lobe suppression. 

These figures show wide variations of both 
phases and amplitudes in individual crystals. 



CRYSTAL ARRAY WITHOUT BACKING PLATE 



23.05 KC 



25.05 KC (RESONANCE) 



27.05 KC 


37.00 KC 



CRYSTAL ARRAY GLUED ON BACKING PLATE 

Figure 20. Velocity distributions of GD28-1 crystal array with and without backing plate. 


probe examination, the individual behavior of 
each crystal can be studied as to its relative ve¬ 
locity, phase, and frequency response. 

The techniques of probe examinations are 
discussed in Chapter 9. Motors are probed both 
in air and in oil. Measurements in air are not 
benefited by the proper impedance loading on 
the motor, but they are easier to make, and they 
reveal vibrational modes which undergo no 
basic change when loaded with liquid. Figure 
20 shows velocity distributions of a motor at 
several frequencies under different conditions. 
The upper group shows the effect of loading one 
side of a crystal array, and the lower group 


Usually, the variations are less when loaded 
with an oil medium, but not always. Variations 
are observed sometimes over the surfaces of in¬ 
dividual crystals. At the resonant points of the 
motor the variations are greatest, probably be¬ 
cause the individual crystal resonances have a 
distribution in frequency. This condition sug¬ 
gests a design that separates the operating and 
resonance frequencies if uniform velocities are 
important. Thus it is not surprising that some 
directivities show erratic behavior, particularly 
in the side lobes, as this nonuniformity would 
control them more than the main lobe (see Sec¬ 
tion 4.3 on directivities). 




























108 


COMPONENT PARTS 


As an example of the control over construc¬ 
tion possible with this technique, the history of 
the GD28 will be cited. The GD28 is a trans¬ 
ducer made up with 96 i/4 by % by 1.2 in. Y-cut 
crystals glued on a 1/2 by 41/2 by 4 in. steel back¬ 
ing plate. Every step of the construction was 
closely controlled, and measurements were 
made on the unit at each step. 

The crystals were first individually measured 
for activity by the probe and by the position of 


in frequency graph, the resonant frequencies 
appear as shown in Figure 23. The maximum 
variations in resonant frequencies ranged from 
45.4 to 47.9 with a mean at 46.80 kc. The mean 
deviation was, however, quite small, being 0.42 
kc which is about 0.9 per cent. The variations 
are, however, larger than those of the unas¬ 
sembled crystals. The relative velocities of each 
crystal at resonance are plotted in a distribution 
in velocity chart shown in Figure 24. The ve- 



0 db -3 db -6 db -10 db -20db 


Figure 21. Velocity and phase distributions of GD28-1 motor in air and in oil at 25.5 kc. 


the discontinuity in the overall impedance 
curve, and only those that were within a 3-db 
limit of activity under constant voltage were 
chosen. The resonances of all crystals were held 
between 46 and 47 kc. They were then assem¬ 
bled into a matrix 8 by 12 crystals, each crystal 
separated from its neighbor with %-in. sponge 
rubber. No glue was used in the construction, 
the matrix being held together loosely with a 
clamp. The resonant frequency of each crystal 
was then measured; relative velocity of each 
crystal at resonance and velocity distributions 
over the face of this motor were measured at 
five frequencies. When plotted in a distribution 


locities vary over a considerably wider range 
than 3 db, the initial tolerances. Figure 20 
shows the surface-velocity distribution of the 
motor at other frequencies. The variations at 
resonances are larger than at other frequencies 
unless the frequencies are high enough to excite 
complicated modes. It seems that in this case, 
at least, the foiling and assembling of 96 crys¬ 
tals into a matrix has contributed to their in¬ 
dividual differences in resonant frequencies and 
sensitivities. 

The crystal matrix was then glued on the 
V 2 -in. steel backing plate with the baked bake- 
lite gluing technique developed in this labora- 
















































































































































































































PROBE EXAMINATION OF MOTORS 


109 


tory (Chapter 8), and the measurements re- percentage distribution of 2 per cent on the 
peated. The charts of resonant frequencies and backing plate is considerably less than 21 per 
velocities at resonance are in Figures 25 and cent off the backing plate. However, the surface- 
26, and the velocity surface distributions at velocity distributions shown on Figure 20 indi- 
four frequencies are shown in Figure 20. The cate wider limits of variations in the velocities 



Figure 22. Velocity distributions of QBF motor in air. 


mean percentage deviations in the resonant 
frequencies with and without backing plate are 
about the same showing that the backing plate 
had little effect in perturbing this distribution. 
The surface-velocity distributions at resonance 
are shown on Figure 26. In this case the mean 


with the plate than without. These large limits 
probably are caused by unequal gluing of the 
crystals. 

The effect of the backing-plate modes on the 
crystal-surface velocities are shown on Figure 
14, Section 3.4. The plate modes chosen were re- 
















































































































































































































































































































no 


COMPONENT PARTS 


moved from the resonance of the crystals far 
enough to be free of complications of that sort. 
Figure 14, Section 3.4, shows the modes of the 
free plate. There are no modes at frequencies 
betiveen the ones shown there. At 21.21, 37.42, 
37.55, and 68.58 kc prominent modes were 
mapped from the back of the backing plate, and 


effect of flexural modes upon the overall re¬ 
sponse of a transducer was encountered in an¬ 
other unit known as the JC2Z1. This unit is 12 
in. in diameter designed to operate at 24 kc with 
a lobe suppression scheme of two velocity zones 
whose velocity ratios were 3 to 1. The response 
curve is shown in Figure 27. At 24 kc the re- 


> 1 I_I-1-1- i 1 —j-1-1 I-1 I-1-1 t » t — I — j-1-1-1- 1 . .< 

45.4 .6 .8 46.0 .2 .4 .6 .8 47.0 .2 .4 .6 .8 .9 

KC 

Figure 23. Distribution of resonant frequencies of GD28-1 motor without backing plate. Mean: 46.80 
kc. Mean deviation: 0.42 kc. 


the corresponding phase distributions of the 
crystal surfaces were mapped. To assure the 
irregular crystal phases at the mode frequencies 
being due to these modes, phase distributions at 
frequencies slightly removed were also mapped 
and shown in the figure. In every case the fre- 


sponse has a 5.5-db dip instead of a peak thus 
rendering the unit useless for its intended serv¬ 
ice. Its directivity pattern at 24 kc was badly 
distorted showing no symmetry or regularity. 
Probe examinations revealed, at this frequency, 
a prominent flexural mode in the backing plate 


2.0 3.0 4jO 5.0 6.0 7.0 8JD 9.0 10.0 lIjO 12.0 

VELOCITY (RELATIVE SCALE) 

Figure 24. Distribution of velocities at resonance in GD28-1 motor without backing plate. Mean: 7.04. 
Mean deviation: 1.48. 


quencies off the modes gave uniform patterns, 
while the mode frequencies gave irregular 
phase patterns. 

Studies of the GD28 reveal in a general way 
many interesting things about the combined 
actions of crystals coupled together through a 
steel plate, but specifically, the data are not com¬ 
plete enough to warrant many general conclu¬ 
sions about its overall performance. A clear-cut 


that had a nodal circle at about the boundary of 
the two velocity areas. The central part was 
thus 180° out of phase with the outer ring. 
Probe examination in air of the velocity ampli¬ 
tude of the inner crystals is shown in Figure 27 
showing a large dip in their activity at 24 kc. 
The crystals of this unit were grouped four to 
the group, and glued on a backing plate of steel 
% in. thick, backed up with an additional layer 



















PROBE EXAMINATION OF MOTORS 


111 


0.638 in. thick of a low-melting metal called 
Cerrobend. For the purpose of separating the 
effects of the longitudinal modes from the flexu¬ 
ral of this backing system upon the motor face, 
individual units consisting of four crystals 


individual crystals, but the frequencies of the 
crystal dips vary from crystal to crystal cover¬ 
ing a region of 21 to 25 kc, thus softening this 
general dip for the transducer. 

Probing technique has not yet been developed 



I_ 

24.0 


_u 

,4 


“F -*-'-'-‘-'-'- 1 -'- 1 - 1 - 1 _> 

25.0 .2 .4 .6 .8 26.0 


Figure 2o. Distribution of resonant frequencies of GD28-1 motor with backing plate. Mean • 25 19 kc 
Mean deviation: 0.177 kc. ’ ‘ 


cemented to a 1-in. sq cross section of this back¬ 
ing system were prepared and probed. The 
general response of such units is shown in 
Figure 27 by the dashed curve. No sharp dip is 


to the point of establishing accurate cause and 
effect relations between surface velocities and 
directivities or calibrations, but many valuable 
uses of it can be made as a tool in controlling 


• ••••••••••••••• ••• 


-* 1_I_I_I-1-1-1_■_I I —j 1 

.07 .08 .09 .1 .2 .3 .4 .6 .6 .7 .8 .9 1.0 

VELOCITY (RELATIVE SCALE) 

Figure 26. Distribution of velocities at resonance in GD28-1 motor with backing plate. Mean: 0.44. 
Mean deviation: 0.01. 


evident, which means that the dip in the motor 
must be due to flexural modes of the large plate. 
The response curve of the transducer in water 
does not have such a sharp dip at 24 kc as do the 


the construction of transducers. A few ex¬ 
amples of its use have been cited which encour¬ 
age its development to refinements that will en¬ 
able it to give more exact information about 
















112 


COMPONENT PARTS 


the inner workings of transducers. Such infor¬ 
mation is necessary before accurate design pro¬ 
cedures can be established. 



Figure 27. Probe studies of the JC2Z-1 trans¬ 
ducer. 


3 7 PARASITIC MODES 

^ Solid Parts 

The transducer case includes all the parts re¬ 
quired to hold the active components together 
and to fulfill the assorted mechanical require¬ 
ments which may be put on a transducer. Thus, 
in certain instances, it may be profitable to re¬ 
gard the mounting structure, a streamlined 
dome, and even the entire hull of a ship as the 
“case.” In general, any part which is not of it¬ 
self an active part but which is or may be acous¬ 
tically coupled to active parts should be re¬ 
garded as part of the transducer case; the de¬ 
gree to which any such passive part affects the 
transducer behavior depends on the degree to 
which it is coupled. At the least, the case con¬ 
sists of whatever barrier separates the crystals 
from the sea water. 

Usually the case includes several metal or 
glass parts: materials of moderately high me¬ 
chanical Q. Also the frequencies are usually 
such that the dimensions are “comparable with 
a wavelength” in these materials. It is to be 
expected that resonances of the case corre¬ 
sponding to various compressional, flexural, 
and torsional vibrations will occur in the de¬ 


sired operating frequency band. Except in very 
rare instances the case structure is so complex 
that calculation of these resonances is impos¬ 
sible; this very complexity probably increases 
the number of resonances which may occur in 
a given frequency band. 

These resonances may be divided into two 
kinds: (1) those in which the vibration leads 
to radiation into the water or other fairly high 
resistance; (2) those in which the vibration is 
only slightly damped. The first kind will have 
relatively low Q and will exert their influence 
over a considerable frequency band; the second 
kind will affect only a very narrow band. The 
line of demarcation is hypothetical; actually 
resonances grading from one extreme to the 
other may be encountered. 

Since resonances of the first kind are damped, 
the impedance into which energy must be put 
to excite them will be relatively high even at 
resonance. For this reason, they usually absorb 
relatively less energy and the effect on the de¬ 
sired frequency-response curve of the trans¬ 
ducer is not great. However, these vibrations 
may radiate significant amounts of energy in 
directions (and phases) at which the primary 
radiating face radiates very little, and it is 
quite likely that the directivity patterns will be 
distorted by this radiation. 

The second kind of resonance radiates no sig¬ 
nificant energy and can influence directivity 
patterns only indirectly. For this reason, it is 
unlikely that such a resonance will affect the 
directivity patterns. Instead, the low impedance 
presented at resonance is likely to absorb a 
large amount of energy which will simply run 
around in the structure and ultimately be ex¬ 
pended in internal losses. The result is usually 
a rather deep hole in the frequency response of 
the transducer. 

Bearing these properties in mind, one has 
certain clues to correcting misbehaviors in a 
transducer. If a sharp hole appears in a fre¬ 
quency-response curve, one is inclined to ques¬ 
tion the behavior of various internal parts 
which do not contact the water; if the direc¬ 
tivity patterns are distorted, one questions the 
behavior of the external structure and of the 
backing plate. Of course, these effects are 
greatly complicated by cavity modes (see Sec- 



































WINDOWS 


113 


tion 3.7.2) and by the various vibrational modes 
which may occur in the active parts (for ex¬ 
ample, see Section 3.4) so that a trial and error 
method is required to find the cause of a given 
misbehavior. 

Since the modes which may occur in a trans¬ 
ducer case are beyond calculation, it is best to 
strive at the outset to avoid exciting any mode. 
This may be done by designing with three rules 
in mind: (1) minimize the number of parts; 
(2) attempt to choose dimensions which are not 
likely to resonate in the operating frequency 
band; (3) incorporate as much isolation as is 
practical so as to decouple the case from the 
active parts. A more detailed discussion of the 
methods of accomplishing this is given in the 
section on design procedures in Chapter 6. 


^ Cavity Modes 

The distinction between cavity modes and 
case modes (Section 3.7.1) is not definite, but 
a rough distinction is profitable. Frequently a 
transducer contains castor oil or some similar 
liquid to couple the crystals to the sea water. In 
one sense, this liquid is part of the case and the 
remarks about case modes are pertinent here 
too. However, there is at least one great dif¬ 
ference, and that is that the liquid cannot be 
wholly decoupled if it is going to fulfill its func¬ 
tion. Thus, one is presented with a mass of 
castor oil, contained in some usually peculiar 
shape, through which vibrations of a particular 
kind must be propagated. Even if the walls of 
the container were perfectly rigid, vibrational 
modes might exist within this cavity. Since the 
boundaries are actually case components, there 
are complicated impedance-boundary condi¬ 
tions imposed which bring in questions of case 
resonances as well as cavity resonances. In gen¬ 
eral, we may crudely visualize vibrational 
modes which are dominated by cavity geometry 
and call them cavity modes. 

Like the case modes, these cavity modes are 
beyond calculation. If the dimensions of the 
cavity become comparable with a wavelength in 
the cavity medium, resonances must be antici¬ 
pated; and the more complicated the cavity 
shape, the greater the likelihood of their being 


troublesome. Such a cavity mode may have any 
Q, depending on the boundary impedances, and 
may or may not radiate into the water. Usually 
the most noticeable effect is on response curves 
rather than on directivity patterns, although 
the latter may be affected. 

If trouble is encountered and a cavity mode 
is suspected, only two courses of action are 
known: (1) drastically change the cavity 
geometry in hopes that some change will be ef¬ 
fective and no new modes will be formed; (2) 
decouple as much as possible by putting foam 
rubber or similar material on the crystals and 
other parts if necessary. Obviously, the first 
method is based wholly on wishful thinking 
unless it is possible to so alter the geometry 
that no dimension remains comparable with a 
wavelength. In that event, this first method is 
preferable; otherwise, the second is more likely 
to work in a small number of attempts. In fact, 
this second method has been so successful that 
it is now common practice to put foam rubber 
in during initial construction. Actually, when 
one puts in foam rubber and it works, one won¬ 
ders forever after what was cured; the cavity 
modes, case modes, etc., cannot be artificially 
divorced from each other. 

Every transducer presents a new problem. 
By experience and intuition one may acquire the 
knack of recognizing which structures are most 
likely to give trouble, but this skill is very 
limited and occasional unpleasant surprises will 
always occur. It is essential that the designer 
be constantly aware of case and cavity modes 
as possible explanations of misbehavior, since 
erroneous diagnosis is very easy. 


38 WINDOWS 

In crystal transducers, the window may 
serve to separate two liquid media, as sea water 
and castor oil, or the crystals may be attached 
directly to one side of the window, the latter 
not only protecting the crystals from the action 
of sea water but also serving as their means of 
support. In either case, an attempt is usually 
made to match the acoustic resistance of the 
window with that of the contiguous liquid 
medium. For a solid window whose shear modu- 



114 


COMPONENT PARTS 


lus is negligible, this reduces to a matching of 
the density q and the acoustic velocity C, or of 
their respective products for the media in¬ 
volved. Treatment of the simple cases of reflec¬ 
tion and transmission of sound waves may be 
found in various texts.The more general case 
of a solid whose shear mode must be taken into 
account and where the waves are incident at 
any angle will be treated in the next paragraph. 

Transmission of Plane Waves by a Plane 
Parallel Solid Windoiv, Neglectmg Attemiation 
Losses. The fraction T of sound energy trans¬ 
mitted by a solid window of density immersed 
in a liquid of density q and velocity C is given 
by the following equation due to Reissner.^^ 


r % - 1). + (93) 

where 

^ ^ Ps cos d r Vd cos- 2dr Vr SiW 2dr 

pC [_ cos dd sin <t> ' cos 6r sin y J’ 

^ Ps cos e r Vd cos“ 2dr Vr sin^ 2Br 1 

pc l_cos0dtan cos^rtany J‘ ^ 

The velocity of the shear wave in the solid 
Vr is equal to Vn/q^, where n is the shear modu¬ 
lus ; the velocity of the longitudinal wave in the 
solid V^ is equal to where E is the 

bulk modulus; 6 is the angle of incidence of the 
wave in the liquid medium, measured from the 
normal. The angles of refraction of the longi¬ 
tudinal and shear waves in the solid, 6„ and 6»,., 
respectively, are determined from Snell’s law 
of refraction, hence: 


Vd 

sin dd = -Q- sin 6, and sin dr 


Vr . 
= -^sm 


(96) 


ing the following equation graphically, one can 
readily obtain that value of d which corre¬ 
sponds to the angle of incidence for zero trans¬ 
mission. 



Lcd 


The usefulness of Reissner’s equation in pre¬ 
dicting the acoustical transmission behavior of 
solids as a function of the angle of incidence is 
illustrated in Figure 28 where Mason’s^^ calcu¬ 
lated and experimental curves for the plastic 
Lucite may be compared. The agreement is ex¬ 
cellent, particularly with respect to the position 
of the sharp minimum at 44.5° from the nor¬ 
mal. 

For solids where the shear modulus is negli¬ 
gibly small, as in some types of rubber, equation 
(93) reduces to: 


T - 


If pc cos dd 
^^LPsVd cos d 


_ p,Vd cosg ~l 
pC cos dd J 

^(^COS9.) 


(99) 


In Figure 28, this function has been plotted for 
a sample of rubber together with experimental 
transmission data, both taken from work by 
Mason.^ Again it will be noted that the agree¬ 
ment as a function of the angle of incidence 
is good. 


The angles 4> and y are defined by the relations: 

iod , ijod 

(f) = jyr cos dd, and y = yy- COS dr, (97) 
V d V r 

where d is the thickness of the window. 

When the angle of incidence exceeds the 
critical angle for the longitudinal and/or the 
shear wave, then d^j and/or d,. become imagi¬ 
nary so that some of the quantities in the terms 
for M and N must be replaced by hyperbolic 
functions. Reference to the original paper of 
Reissner^i should be made for the modifications 
required in these critical regions. 

An angle of zero transmission always occurs 
in the neighborhood of a critical angle. By solv¬ 


Rubber 

Rubber has been widely adopted for acoustic 
windows in transducers, primarily because of 
the good impedance match obtainable with sea 
water but in part due to its elastic properties, 
abrasion resistance, and its electrical resis¬ 
tivity. It will be noted in Table 4 that either q 
and c, or the product qc, for sea water may be 
quite closely duplicated in rubber by variations 
in its composition. Where both q and c are 
matched, it is customary to refer to the rubber 
as “qc” rubber. The effective magnitude of the 
acoustic velocity in a solid body depends on its 
shape. In a long rod, it is found that the velocity 

















INSERTION LOSS IN DB INSERTION LOSS IN DB 


WINDOWS 


115 



-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 


ANGLE FROM NORMAL IN DEGREES 

Figure 28. Transmission loss as a function of angle of incidence for single rubber and Lucite plates, 
under stated conditions. (Bell Telephone Laboratories.) 































































116 


COMPONENT PARTS 


is determined by the value of Young’s modulus. 
For example, most handbook values for the ve¬ 
locity of sound in rubber have been based on 
measurements of rodlike samples and usually 
range from 30 to 50 m per sec. For sound 
windows, the rubber is usually employed in 
thick sheets rather than as a rod, hence the 
velocity values in Table 4 are the proper ones 
to use and range from 1,500 to 1,600 m per sec. 
These latter values depend on the shear and 
bulk moduli of the solid. For “sound” rubber. 


excellent. A somewhat similar effect was ob¬ 
served with a plano-convex neoprene window 
where the initially unsatisfactory directivity 
patterns of the transducer were much improved 
later by planing off the convex surface of the 
rubber. 

To increase the mechanical strength of rub¬ 
ber windows, they may be reinforced with steel 
bars, as shown elsewhere in this book for the 
GD34Z-1 transducer. These bars do not inter¬ 
fere with the radiation in that particular case 


Table 4. Materials of construction for sound windows and their various constants of 
acoustic importance.* 




Density 

Velocity 

Acoustic 

Bulk 

Shear 


Loading 

(o) 

(c) 

resistance 

modulus 

modulus 

Material 

factor 

g/cm3 

m/sec 

(qc) 

{E) dynes/cm2 

(r) dynes'cm2 

Sea water 


1.03 

1,500 

1,545 

2.36 X 1010 

0 

Castor oil 


0.95 

1,540 

1,460 

2.51 X 1010 

0 

Steel 


7.00 

5,000 

35,000 

179 X 101" 

7.58 X 1011 

Sound rubber 

0 

1.03 

1,530 

1,575 

2.38 X 1010 

2.76 X 106 

Tire-tread rubber 

50% C. 

1.15 

1,600 

1,840 

2.94 X 1010 

27.6 X 100 

Hycar Os-30 


0.96 

1,560 

1,500 

2.34 X lOio 

9 

Rubber 

10:100 ZnO 

1.03 

1,530 

1,580 

2.48 X lO’o 

? 

Perbunan 


1.12 

1,650 

1,850 

3.05 X 1010 

9 

Butyl rubber 


0.97 

1,630 

1,580 

2.58 X 1010 

9 

Koroseal 

62.5:100 TCP 

1.30 

2,160 

2,810 

6.15 X lOio 

9 

Neoprene G 


1.32 

1,500 

1,980 

2.96 X 1010 

? 

Lucite 

1 

1.185 

1,981-2,002 

2,360 

9 

9 

Type 8388 (Goodrich) 

9 

1.15 

1,550 

17.8 

9 

9 

Type 79-SR-32 

9 

0.99 

1,525 

15.1 

9 

9 

(Goodrich) 








* Courtesy of Goodrich Rubber Company, Naval Research Laboratories, Bell Telephone Laboratories. 


the shear modulus is small and may be neg¬ 
lected; in other types of rubber, it may be im¬ 
portant in some cases. 

As long as acoustic radiation is incident ap¬ 
proximately normally on a rubber window, an 
accurate match to the qc of sea water appears 
quite unnecessary. For large angles of inci¬ 
dence, however, refraction effects have been 
found troublesome. In the case of a cylindrical 
rubber sleeve surrounding a square array of 
crystals, used as a motor, where the rubber 
stock was neoprene of approximately 40 Shore 
durometer and may have possessed an appre¬ 
ciable shear modulus, the large variations in the 
angle of incidence upon the rubber sleeve for 
the same polar angle subtended at the crystal 
motor resulted in a very distorted directivity 
pattern. When sound rubber (oc) was used 
with the same crystal motor, the patterns were 


since the crystals are attached to the rubber in 
locations where they may radiate freely be¬ 
tween the steel bars. The window for the 
GD34Z-1 withstood a pressure test of 250 psi. 

Where less strengthening is desired, the rub¬ 
ber may be molded over expanded metal screen 
or reinforced with hardware cloth and still re¬ 
tain its valuable acoustic properties. An impor¬ 
tant point is to guard against entrapped air 
during the molding or forming operations. By 
resorting to “press” curing, air-free stocks can 
be consistently obtained. The presence of oc¬ 
cluded air, of course, would lead to very low 
transmission. 

Lucite 

Data on the acoustic properties of Lucite by 
Mason^ indicate that it may be useful over cer¬ 
tain frequency ranges, but the variation of its 















WINDOWS 


117 


constants with frequency render it unusable in 
other frequency ranges. The normal loss 
through %- and %-in. Lucite sheets as a func¬ 
tion of frequency is shown in Figure 29. The loss 
is small below 40 kc and in the immediate neigh¬ 
borhood of 60 kc; at 47 and 75 kc there are 
maxima in the loss curve. The position in the 



10 20 30 40 50 60 70 80 90 100 MO 120 130 140 

FREQUENCY IN KC 

Figure 29. Transmission loss at normal in¬ 
cidence as a function of frequency for two 
Lucite plates, % and % in. thick. (Bell Tele¬ 
phone Laboratories.) 

maxima appear to be independent of the thick¬ 
ness. 

The velocity fluctuates between extreme val¬ 
ues of 1,981 and 2,002 m per sec in the 20- to 
110-kc range, the two maxima appearing at 43 
and 71 kc. 

Tenite 

The cellulose acetate butyrate plastic known 
as Tenite II has a lower acoustic velocity than 
Lucite and hence larger angles of incidence for 
zero transmission. The loss curve at normal in¬ 
cidence for Tenite II-H5 shows a single maxi¬ 
mum at 37 kc of 3 db for a Y-z-in. sheet; above 
50 or below 28 kc, the loss is only 1 db or less. 
For the range 20 to 30 kc, the best grade ap¬ 
pears to be H-4. Mason,1 from whose work the 
above information is obtained, states that good 
transmission is also obtained if Tenite II is 
molded about an expanded-metal supporting 
structure. 


Nylon 

Mason has also measured nylon in the 20- to 
30-kc range and found the normal loss to be 
very low. The angle of zero transmission for 
nylon is around 62 degrees. 

Steel 

The transmission behavior of a i/4-in. steel 
sheet when used as a window between sea water 
and a second medium having the same acoustic 
constants as water has been calculated on the 
basis of equation (93) for a frequency of 60 
kc. The assumed values for the velocity of the 



Figure 30. Plate separation as a function of 
frequency for perfect transmission at normal 
incidence through pairs of steel plates of given 
thickness. 

longitudinal and shear waves were 5.75 X 10^ 
cm per sec and 3.12 X 10^ cm per sec, respec¬ 
tively. The transmission is quite uniform for 
angles of incidence up to 10° from the normal 
and is approximately 3 per cent. The first angle 
of incidence for zero transmission occurs near 













































































118 


COMPONENT PARTS 


17°. The transmission behavior in the critical 
regions is quite irregular. 

The transmission loss through a 0.030-in. 
stainless steel covering on a 54-in. dome has 
been measured by Dietze.^^ His experimental 
values for frequencies of 10, 20, 25, 30, 50, and 
60 kc are 0.4, 0.7, 0.3, 0.9, 1.6, and 1.8 db, respec¬ 
tively. In addition to the use of thin stainless 
steel for the construction of domes, thin tinned- 
steel coverings have also been used to house 
transducers. In the “tin-can” transducer case 
discussed in Chapter 8, the wall thickness is 
approximately 0.010 in. and hence would have 
a transmission loss of roughly one-third to one- 
fourth of the values given above. As the thick¬ 
ness of a steel plate is increased, the transmis¬ 
sion loss becomes greater and greater until the 
plate is no longer of value as a window. How¬ 
ever, the transmission does pass through a 
minimum at some particular thickness and then 
increases to 100 per cent again when the plate 
thickness becomes equal to a half wavelength 
of the radiation being used. Although half-wave 
plates may find some application as a sort of 
directional filter, the rapid decrease in trans¬ 
mission as one departs from a normal angle of 
incidence is too rapid and critical for most uses. 
Their transmission is also a critical function of 
frequency. 

A better solution to the problem of transmit¬ 
ting acoustic energy through a steel window 
having considerable strength consists of using 
two comparatively thin steel plates separated 
by a layer of material which has the same 
acoustic impedance as the liquid on the outside 
of the two steel plates. The theory for the trans¬ 
mission through such a double-layer wall has 
been developed by McMillan.The theory used 
in Section 3.5 should also be applicable here. 
A graph based on the equations of McMillan is 
given in Figure 30, and shows the plate separa¬ 
tion for perfect transmission through a pair of 
plates as a function of frequency for the case of 
steel plates having thicknesses of Yic,, Vn, and 
1/4 in. The angular variation in transmission 
through a pair of Mr.-in. steel plates, with sepa¬ 
rations of 0.064 and 0.072 in. respectively, for 
a frequency of 70 kc, has been computed by Mc¬ 
Millan and is reproduced in Table 5. The results 
in this table indicate that good transmission at 


70 kc can be obtained over a wide angular range 
through a window having considerable me¬ 
chanical strength. 


Table 5. Per cent transmission as a function of 
angle of incidence through a pair of parallel steel 
plates, plate thickness i/i« in., frequency 70 kc. 


Angle of 

Per cent transmission 

incidence 

0.064-in. plate 

0.072-in. plate 


separation 

separation 


0° 

100 

82 

10° 

99 

89 

20° 

86 

100 

30° 

68 

83 

45° 

37 

44 

60° 

33 

36 


To test the performance of a window possess¬ 
ing this double-layer construction, a number of 
experiments have been conducted at UCDWR. 
Tests were made on a pair of %o-in. thick steel 
plates separated by distances of 0.044 and 0.143 
in., respectively. These pairs of plates were sup¬ 
ported immediately in front of appropriate 
flat-faced transducers and the responses meas¬ 
ured as a function of frequency both with and 
without the various sets of plates. At those fre¬ 
quencies where the theory indicated perfect 
transmission for the respective double-layer 
plates, it was found that the measured response 
through them was equal or slightly greater than 
without them. These regions of high transmis¬ 
sion were quite sharp, but would be suitable for 
transducers which operate at a single fre¬ 
quency. The transmission behavior at other 
frequencies was not always completely under¬ 
stood. One would expect difficulties such as 
ffexural modes to occur in these plates as they 
do elsewhere in steel plates of comparable thick¬ 
ness. The directivity patterns at the frequency 
of maximum transmission through this type of 
window were usually as good as, or slightly 
better than, without them. At frequencies a few 
kc higher or lower than the frequency of maxi¬ 
mum transmission, however, the directivity pat¬ 
terns were often greatly distorted. This would 
seem to limit the usefulness of this type of con¬ 
struction to transducers operating over a rather 
narrow frequency range which centered about 
the frequency of maximum transmission. 








PROPERTIES OF INERT TRANSDUCER MATERIALS 


119 


The strength obtainable in such a steel sand¬ 
wich window would suggest its use in sonar 
domes. Several experiments on cylindrical 
domes were conducted to test this point, one of 
them being on a pair of i/4-in. wall concentric 
cylinders about 10 in. in diameter, another on 
a pair of concentric cylinders having 
wall and a diameter of 18 in. In both cases it 
was found that the directivity patterns were 
very badly distorted, apparently owing to in¬ 
ternal reflections of radiation incident at angles 
other than normal. Since domes should give 
little or no interference to the directional char¬ 
acteristics of transducers, the use of a double¬ 
layer type of construction is only suggested for 
this purpose when the radius of curvature can 
be made quite large. However, it should be 
pointed out that these experiments were pre¬ 
liminary and exploratory in nature and a more 
profound investigation might still develop 
something useful for particular applications. 

An ultimate usefulness may develop for this 
sandwich-type plate in furnishing a strong win¬ 
dow for transducers in weapons which are 
either rocket fired or airplane launched. In this 
case they would strike the water at very high 
velocities. Even then they would probably have 
to be designed to operate at a single frequency 
or over a very limited range. 


PROPERTIES OF INERT TRANSDUCER 
MATERIALS 

A transducer of any size and complexity usu¬ 
ally contains a great many parts besides crys¬ 
tals and backing plates. Usually parts are re¬ 
quired to perform such functions as: support 
the active parts within the case, provide electric 
connections, support wires, convey wires 
through watertight bulkheads, provide acoustic 
isolation, attach various parts to each other, 
provide a strong container, and provide an 
acoustically transparent window. At one time 
or another a great many materials have been 
used to perform these various functions, and 
the following is a brief outline of salient proper¬ 
ties of the more important substances. The nu¬ 
merical values given are the best available; 
most of them are taken from the standard ref¬ 


erence literature, but the sources are so diverse 
that no attempt is made to make proper 
acknowledgment. 

Metals 

Aluminum (q = 2.7; c = 5.1 X 10’' cgs 
units). Aluminum has not been used ex¬ 
tensively for backing plates because of the 
relatively great thickness required for a high 
impedance. The mechanical Q is quite high. 
Aluminum does not solder or weld easily and is 
corroded rapidly in sea water. Aluminum alloys 
might be quite useful for passive internal parts 
because of light weight. 

Brass (various) (67 Cu, 33 Zn) (q == 8.40; 
c = 3.5 X lO"’). Brass has not been used for 
backing plates and there is no apparent advan¬ 
tage thereto. Brass behaves like copper in the 
presence of castor oil and RS, but small brass 
parts frequently have been used, generally 
when cadmium or nickel plated. Brass solders 
and machines well, and stands up reasonably 
well in sea water. Some trouble may be en¬ 
countered from bimetallic corrosion. 

Bronze (various) (q, c vary hut similar to 
copper). Bronze is used only for cast exterior 
cases; in this service the bronze may be tin 
dipped and works very well. However, such a 
case is quite heavy and expensive. Bronzes gen¬ 
erally behave like copper in the presence of 
castor oil and RS but may be protected by plat¬ 
ing. The wide range of commercial bronzes 
offers many interesting properties. Silicon, 
phosphorus, and beryllium bronzes have occa¬ 
sional specialized applications such as in 
springs. 

Cadmium. Cadmium is useful only as a plat¬ 
ing on other metals. On steel it affords some 
protection from sea water, but it cannot be re¬ 
lied upon. Usually it will rust through in a week 
or two. Experience indicates that cadmium af¬ 
fords considerably better protection to steel 
than do nickel and chromium but not as good 
as a heavy tin dip or plate. There appears to be 
no metallic spray, plate, or dip which reliably 
protects iron from sea water. 

Copper (q = 8.5; c = 3.56 X 10“). Copper has 
not been used for backing plates, although it 
might be useful in particular instances. Copper, 
RS, and castor oil are reputedly intolerant, al- 




120 


COMPONENT PARTS 


though any pair are tolerant. The trouble comes 
from the formation, over an extended time, of 
a black conducting sludge in the castor oil asso¬ 
ciated with corrosion of the copper. The copper 
may be protected by heavy plating. No data are 
available on copper, castor oil, and ADP but the 
combination has been used with no evidence of 
trouble within one year. Copper may be used 
for internal electric wiring, but the high crys¬ 
tal impedance usually allows the use of less 
questionable metals, such as iron; any copper 
wire so used must be tinned when in the pres¬ 
ence of castor oil and RS. The work-hardening 
of copper during repeated flexure may lead to 
broken wires if a transducer is subject to vibra¬ 
tion. Since any transducer aboard ship may be 
subject to such vibrations, any wire should be 
thoroughly secured (with mechanical resist¬ 
ance if possible) to avoid resonant vibrations. 

Gold. As an evaporated layer, gold has been 
used as electrode attached to crystal. In foil, or 
plated on other foil, it has been used to make 
connections to crystal electrodes. Evaporated 
gold electrodes are advantageous for 45° X-cut 
RS, but this is not required for 45° Y-cut RS 
or 45° X-cut ADP, except in very high-power 
transducers. 

/ron (cast white) (o = 7.6; c = 5.1 X lO'A. 
Ordinary cast iron has been used for many 
transducer parts, particularly exterior cases 
and backing plates. It is quite brittle and very 
difficult to solder or weld. One generally has 
difficulty getting castings which are not porous. 
There may be advantage in the fact that cast 
iron is quite stable against cold flow and other 
inelastic deformation. Cast Meehanite is su¬ 
perior to ordinary cast iron in many ways, 
chiefly for the chance of getting good castings, 
and for superior strength, Meehanite is a pat¬ 
ented process, licensed to various foundries 
(information from Meehanite Corp.). Cast 
steel is probably preferable to either of the 
above, but has been used very little. Hot- and 
cold-rolled steel are the most common materials 
in transducers. Hot and cold forgings are usu¬ 
ally more suitable for production quantities 
than for the small quantities dealt with in ini¬ 
tial design. For this reason, the main compo¬ 
nents of many Submarine Signal Company 
projectors (to name only one manufacturer) 


are forgings. In general, iron and steel are as 
good a material as any in exposure to sea water, 
provided wall thicknesses are sufficient to allow 
some rusting. Nuts and bolts must be protected 
from rusting on both heads and threads. Nearly 
all UCDWR backing plates have been cast 
Meehanite or hot- or cold-rolled steel. When 
vitreous porcelain enamel is to be applied, the 
enameling concern should be consulted about 
impurities which may be harmful to the enamel. 
There is no reason to think that the black scale 
occurring on hot rolled and castings is harmful 
to transducer interiors, provided the scale can¬ 
not fall off. 

Various surface treatments such as dips and 
platings can be readily applied to steel but af¬ 
ford only partial corrosion protection, and in 
some cases are harmful because of bimetallic 
corrosion. The most common methods of fabri¬ 
cation are soft and hard soldering and welding. 
The soft solders are suitable only for fairly 
small parts where strength is not a require¬ 
ment. The hard solders are suitable only for 
fairly small parts which can be properly heated 
and fluxed (unless a furnace is available). Gas 
and arc welding are most suitable for strength 
members, and electric arc welding is usually 
easier to do on very heavy parts. Trouble may 
be encountered with welds which leak, but a 
skilled arc welder can reliably make absolutely 
tight welds. Considerable success has been had 
with furnace copper brazing (in special ovens 
with controlled atmosphere), and this method 
is strongly recommended wherever possible. A 
copper-brazed joint should probably be pro¬ 
tected from sea water because of possible bi¬ 
metallic corrosion. UCDWR has generally used 
small (No. 27 AWG) tinned iron wire for elec¬ 
trical hookups within transducers; this wire is 
soft, cheap, easily soldered, and quite strong. 
Nuts, bolts, screws, etc., in the interior are usu¬ 
ally unplated steel. Very thin-tinned sheet iron 
such as commonly used in tin cans has been 
very successfully used in small transducers; it 
can be used for both the case and the acoustic 
window. Stainless steel is rather difficult to ma¬ 
chine and fabricate, but it can be spot welded, 
welded, and soldered if the proper methods are 
used. Besides the obvious corrosion resistance, 
stainless steel has the advantage that marine 




PROPERTIES OF INERT TRANSDUCER MATERIALS 


121 


life, such as barnacles, grow on it only slowly 
if at all. For this reason, also, stainless 
steel is particularly suited for use in thin 
sheets (20 gauge or thinner) as the acoustic 
window. 

Germcm Silver. UCDWR has used large quan¬ 
tities of 0.001-in. German silver foil for making 
connections among crystal electrodes. This foil 
cuts and solders easily and is quite strong. Its 
cost is quite high because of the rolling opera¬ 
tion, and it probably offers no advantages over 
silver foil. There are indications that some cor¬ 
rosion process may occur in contact with RS 
and castor oil (possibly because of the copper 
content) but UCDWR has not had enough trou¬ 
ble to warrant investigation. 

Lead (q = 11.3; c = 1.2 X lO"’). Lead has 
been used as the second layer (with steel) in a 
two-layer backing plate. Because of the low 
value of c, a clamping lead backing plate at 
some frequency is both thinner and lighter than 
a clamping steel plate at that frequency. The 
mechanical Q of lead is considerably lower than 
that of steel, but is still so high as to present 
negligible losses. Lead wool might be used, with 
castor oil, to form a sound-absorbing medium, 
but lead wool crumbles and packs so much 
that its use is of questionable value. Lead is 
occasionally used as a gasket, but this is only 
possible when springs, lock washers, etc., are 
put under the screws to provide constant pres¬ 
sure. If screws alone are used, the lead will 
cold flow to relieve the pressure, and the gasket 
may leak. 

Magnesium. Magnesium or magnesium alloys 
might offer possibilities as backing-plate mate¬ 
rials or internal passive parts; good castings 
are relatively easy when the foundry is prop¬ 
erly equipped. Extreme corrosion prevents its 
prolonged use in contact with sea water. Mate¬ 
rial is now available from Dow Chemical Com¬ 
pany by which magnesium is readily soldered. 

Silver. Silver wire and silver foil have been 
widely used in electric connections to crystals. 
Silver solders easily and is an excellent electric 
and thermal conductor. The price difference be¬ 
tween silver and baser metals is usually trivial. 
Sometimes silver foil will not solder because 
of lubricant forced into it in rolling; this diffi¬ 
culty should not arise and is easily corrected. 


There is no evidence that silver acts like copper 
in the presence of castor oil and RS. 

Tin. The cost of tin precludes its use except 
in thin protecting layers on other metals. Ex¬ 
perience indicates that where a heavy, well- 
bonded tin dip or tin plate can be attached to 
steel it offers the best corrosion protection of 
any to be had by metallic plating. 

Wood's Metal. This alloy and other similar 
low-melting alloys offer possibilties in making 
up two-layer backing plates. These alloys wet 
steel better than pure lead, and the lower melt¬ 
ing point is a definite advantage when attaching 
to the back of a porcelainized steel plate. One 
of the commercial alloys of this type, Cerro- 
bend (o = 11; c = 1.74 X 10''), has been used 
for this purpose. 

Zinc. There is no reason to think that zinc 
offers any particular advantages in backing 
plates, although it would probably be satisfac¬ 
tory. Extensive use has been made of high- 
zinc alloys (particularly the commercial alloy, 
Zamac) for small castings. These alloys ma¬ 
chine easily and cast well, although an excess 
must be poured because of great shrinkage at 
the top surface. The outstanding advantage is 
that these alloys can be handled easily in a small 
laboratory such as is likely to be interested in 
transducer development. 

Plastics 

Bakelite. This plastic is commonly used with 
various fillers, prominent among which are 
fabric, paper, and minerals. The fabric-base 
bakelite has been extensively used for various 
small parts inside transducers. It machines rea¬ 
sonably well and is quite strong. Since it is 
thermosetting, spontaneous cold flow is not as 
pronounced as is some thermoplastics. The ma¬ 
terial has a “grain” and the strength differs 
markedly with direction. The chief disadvan¬ 
tage is the bad behavior under electric break¬ 
down. In the direction of the grain the dielectric 
strength is not great, and if a spark jumps, the 
path is likely to carbonize. For this reason it is 
customary to impregnate in molten ceresin 
(until boiling ceases) or Glyptal (preierably in 
vacuum). Fabric-base bakelite contains a dye 
which colors castor oil yellow-brown but this 
is not at all harmful. Paper-base bakelite can 




122 


COMPONENT PARTS 


be machined to a better finish than can fabric 
base, but seems not to be as strong. It, too, 
shows grain; its dielectric strength is somewhat 
better. Mineral base is quite hard, and can be 
machined and lapped to a high finish. Dielectric 
strength is high, but the material is rather 
brittle. 

Lucite; Plexiglas. Although a difference does 
exist, no distinction need be made between these 
plastics. They have fair electrical properties, 
high dielectric strength, and little or no evi¬ 
dence of “grain.” Since they are thermoplastics, 
they are very likely to undergo spontaneous 
cold flow to an annoying extent. This makes 
them particularly unsuited for packing gland 
parts, etc. When screwed or riveted together, 
pieces are likely to become loose because of cold 
flow. Their great usefulness is for high-voltage 
insulation where power factor and dielectric 
constant are not important. Lucite (14 or 14 in. 
thick) has been used for fronting plates for 
inertia driven transducers; this seems to work 
well, but the design was not pursued beyond 
small units for general laboratory use because 
of the fragility. The specific acoustic imped¬ 
ance of Lucite is a function of frequency and 
undoubtedly of temperature; generally, it is 
quite close to that of water and in some appli¬ 
cations makes a good acoustic window. At an¬ 
gles of incidence in the vicinity of 60°, Lucite 
shows an anomalously high reflection coefficient 
which must be borne in mind. Like polystyrene 
and many other plastics, the attenuation (which 
is a function of frequency) is quite high, and 
very thick slabs might cause excessive loss on 
transmission. Lucite and Plexiglas machine 
well, although they have a tendency to melt, 
may be given an excellent polish, and cement 
readily. 

Polystyrene. For transducer purposes, poly¬ 
styrene is superior to Lucite and Plexiglas. 
Dielectric losses and dielectric constant are ex¬ 
ceptionally low. 

Polythene (Polyethylene). This plastic be¬ 
came available late in World War II and has 
found only limited use in transducers. However, 
its properties are outstanding. Polythene is the 
only other plastic whose electrical properties 
compare favorably with polystyrene. The spe¬ 
cific gravity is less than unity. Polythene is 


rather horny, flexible, and burns slowly. It is 
thermoplastic, and may be formed readily with¬ 
out elaborate facilities. The only known use in 
transducers to date is as a molded thin case 
around a cylindrical low-frequency magneto- 
strictive transducer developed by UCDWR; this 
application was highly successful. 

Many more plastics with a wide variation of 
interesting properties are worthy of comment. 
Among these are Catalin, Lumarith, Laminae, 
Durez, Tenite, Saran, Viscoloid, Vistanex, 
Nylon, Celluloid, and Cellophane. It is strongly 
urged that anyone interested in transducer de¬ 
sign investigate thoroughly the various plastics 
available to him.^^ 

Glasses and Ceramics 

Glass. There are, of course, many kinds of 
glass with quite different properties. Usually 
the density varies from 2.4 to 2.8 and the veloc¬ 
ity of sound from 5 to 6 X 104 The mechanical 
Q is quite high, and glasses are generally well- 
suited for backing plates (such as Brush C-26). 
UCDWR has made a few glass fronting plates 
with indications that this is worth further in¬ 
vestigation. Glasses have the distinct advantage 
that their properties are relatively independent 
of time and temperature. In cementing it is 
often advisable to grind a matte surface on the 
glass to give a “tooth” for the cement. 

Porcelam Enamel. UCDWR has made exten¬ 
sive use of porcelain enamel as an insulating 
layer bonded to a steel backing plate. The exact 
composition is usually a trade secret of each 
concern and no attempt has been made to learn 
it. This is a glasslike substance, usually black 
or white, moderately homogeneous, and usually 
applied to bathtubs, stoves, kitchenware, etc. 
Two general processes are used: the wet and 
dry processes. UCDWR prefers slightly the dry 
process in which a dry frit is sprinkled onto 
red-hot steel and returned to the furnace until 
melted. The preference may arise from the par¬ 
ticular concerns available. Both kinds tend to 
have internal bubbles when the glaze is ground 
off; thus Glyptal impregnation is suggested 
when the porcelain is to be subjected to any 
considerable voltage. Use of snow-white enamel 
greatly aids the examination of cement joints. 

Porcelain. There are innumerable porcelains 





PROPERTIES OF INERT TRANSDUCER MATERIALS 


123 


which have possible uses for electrical parts in 
transducers. One practice worth mention is that 
of the Bell Telephone Laboratories. At one time 
they customarily used thin ceramic wafers ce¬ 
mented between the crystals and the backing 
plate, analogous to the bonded porcelain enamel 
above. This method seemed satisfactory, but the 
porcelain has since been replaced by plastic 
wafers. 

Metal-to-Glass and Porcelain-to-Glass Seals. 
Recently these seals have become available in 
large quantities from such manufacturers as 
Stupakoff, Spertie, Westinghouse, and Corning. 
Within their limitations, these seals are an 
enormous help in bringing electric leads through 
bulkheads. Some difficulties which may be en¬ 
countered are: fragility, inability to stand high 
pressure (great ocean depth), low electrical re¬ 
sistance caused by occasional poor glass. 

Rubbers 

Sound-Water (oc). This rubber, made by 
B. F. Goodrich Company, was developed spe¬ 
cifically for underwater sound transducers. Its 
specific acoustic impedance is equal to that of 
water; the composition is not known. This rub¬ 
ber is yellow, resembling pure gum and is soft. 
It can be bonded to steel and other metals to 
form acoustic windows, and it is readily avail¬ 
able in sheets and hollow cylinders. The rubber 
seems to contain an oil, and for this reason it 
is sometimes difficult to cement to it. While it 
appears to be quite elastic, it is torn rather 
easily, and is somewhat poor mechanically. It is 
slightly crazed by prolonged sea water immer¬ 
sion, but stands up very well. 

Natural Rubber. Because of World War II, 
very little natural rubber other than Sound- 
Water rubber has been used. We are told that 
natural rubber is damaged by sea water unless 
first subjected to a deproteinizing process. 

Synthetic Rubber. Although there are many 
other synthetic rubbers UCDWR experience is 
limited to neoprene. When used in reasonably 
thin plane windows parallel to plane motors the 
effects arising from the slight impedance mis¬ 
match are not serious. However, thick curved 
windows may seriously distort directivity pat¬ 
terns and cause marked irregularities in fre¬ 
quency response. For such service oc rubber 


appears entirely satisfactory. 

The following composition is one often used 
with success by UCDWR. 


Neoprene 
Carbon black 
Rope seed oil 
Zinc oxide 
Compounding 
Shore hardness 


38 per cent 
40 per cent 
10 per cent 
2 per cent 
10 per cent 
55 to 65: A scale. 


Latex and Rubber Suspensions. Several dip¬ 
ping compounds are available by which a thin 
layer of rubber may be applied to various sur¬ 
faces. Many of these contain water and are 
suitable for making streamlined shapes, etc., 
but may not be put in contact with crystals. 
These seem to be porous, and the d-c resistance 
may drop after submersion. There are dipping 
compounds (whose solvent is carbon bisulphide 
or other organic liquids) which do very well 
for applying rubber directly to crystal surfaces. 
Most of these dipping compounds are air drying 
and require no further curing; others require 
subsequent baking at assorted temperatures 
and times. 


Cements 

Bakelite Cement BC6052. This cement, a cy- 
clized rubber, is by far the most commonly used 
cement for attaching crystals to each other and 
to other surfaces. The technique is not com¬ 
plicated, but is tricky. Results are moderately 
good, and its use arises from the fact that it is 
the most practicable cement for RS. It is a com¬ 
mon misapprehension that a bakelite-cement 
joint can be judged by inspection; one of the 
difficulties of the substance is that good and bad 
joints look nearly identical. Solvents are, to 
name a few, carbon tetrachloride, benzene, 
benzine, acetone, toluene, amyl acetate, ethyl 
acetate, and ether. 

Vulcalock. This is a B. F. Goodrich Company 
product. It differs from BC6052, if at all, only 
in having a higher solvent density. 

Cycle-Weld. There are several Cycle-Weld ce¬ 
ments having various uses. UCDWR has con¬ 
siderable success Cycle-Welding ADP to neo¬ 
prene and QC rubber. The method consists in 
the application and curing of a priming coat of 
55-6 cement on the rubber and the bonding of 
the crystals to this by another Cycle-Weld ce- 



124 


COMPONENT PARTS 


ment called C-3. Cycle-Weld cement is a pat¬ 
ented product of the Chrysler Corporation. 

Shellac: Sealing Wax. Since sealing wax is 
predominantly shellac, it behaves quite simi¬ 
larly; it is a little easier to work and comes in 
convenient stick form. Sealing wax is quite soft 
at 100 C and is fluid at a little higher temper¬ 
ature. ADP is not damaged by these temper¬ 
atures, and may be attached to other surfaces 
with sealing wax. The simplest technique is to 
warm the other surface, such as a backing plate 
and spread over it a uniform layer; the crystals 
are then set in place and a slight pressure ap¬ 
plied to thin the layer of sealing wax between 
the surfaces. On cooling, a very strong uniform 
high Q joint is formed, which improves for the 
next day or two and then remains quite stable. 
Since shellac polymerizes at elevated temper¬ 
atures, and even at room temperature, the 
properties of the shellac may depend upon pre¬ 
vious history, times and temperatures during 
fabrication, and subsequent history. There is 
some question whether a transducer so con¬ 
structed might change its properties over a 
span of years or over the temperature ranges 
encountered in various seas. Cement joints 
made this way have quite high Q but are rather 
brittle and thus have only limited use. Chief 
suggested use is in research where speed is a 
convenience. Common solvents are acetone, 
toluene, benzene, amyl acetate, ethyl acetate, 
ether, and chloroform. 

Beesivax and Rosin. This cement, or one using 
other waxes such as paraffin, has the virtues 
that it sticks well to cold surfaces, and its hard¬ 
ness and softening point may be varied by com¬ 
position. An approximately 50-50 mixture melts 
at a temperature just slightly above the de¬ 
struction temperature of RS. The mixture tends 
to supercool so that it may be used as a cement 
for RS using a technique similar to that for 
sealing wax and ADP. The resulting joints are 
quite good after aging a day or so. It is possible 
that a microscopic layer of RS is damaged, but 
this could have no effect except a possible low¬ 
ering of electrical breakdown voltage. The con¬ 
stitution of a mixture of wax and rosin varies 
on successive heatings so that occasional checks 
on melting point are advised. There is no indi¬ 
cation that the cement is harmed by castor oil. 


Common solvents are acetone, toluene, benzene, 
amyl acetate, ethyl acetate, ether, and chloro¬ 
form. 

Rochelle Salt. Molten RS has been used as a 
cement for both RS and ADP crystals. On heat¬ 
ing, RS decomposes into potassium tartrate and 
sodium tartrate (mixture) with the evolution 
of 1/2 mole of water. The molten liquid will re¬ 
crystallize (not as RS), and tends to supercool. 
The resulting joint undoubtedly contains excess 
water, the amount depending on its molten his¬ 
tory. The most successful joints have been made 
using salt which has been molten and exposed 
to air for several hours. There is some indica¬ 
tion that electrical breakdown occurs at a no¬ 
ticeably lower voltage through these joints, 
perhaps because of excess water and perhaps 
because of the properties of potassium and 
sodium tartrates. Such joints have very high Q 
and are somewhat brittle. There is some ques¬ 
tion whether a joint between crystal and steel, 
for example, would crack because of differential 
thermal expansion. 

Acryloid, Ferdico Marine, Ferdico Aviation, 
and Carnauba wax are other cements which 
have been tested. They all have one or another 
disadvantage with no particular advantage. 
Efforts have been made to find a polymerization- 
type cement suitable for cementing crystals, 
particularly one which will polymerize in rea¬ 
sonable time at room temperature. To date, this 
remains a will-o’-the-wisp, although Bell Tele¬ 
phone Laboratories has recently announced a 
new cement. Butyl C, which may be of this type. 

Liquids 

Castor Oil. Baker and Company’s DB-grade 
castor oil remains the most universally used 
liquid in transducers. This oil is rather viscous 
and the odor is milder than is usually associated 
with castor oil. This grade is quite pure and is 
intended for electrical uses. It usually contains 
large quantities of dissolved air and water 
which must be removed in vacuum. The viscos¬ 
ity is a marked function of temperature and the 
coefficient of thermal expansion is rather large. 
Castor oil becomes turbid at 12 C. The density 
at room temperature is 0.96 to 0.97; the specific 
acoustic impedance at room temperature is very 
close to that of water (the temperature depend- 




PROPERTIES OF INERT TRANSDUCER MATERIALS 


125 


ence of the specific acoustic impedance of castor 
oil resembles that of most other liquids; sea 
water is anomalous and has slope of opposite 
sign). Castor oil does not harm natural or syn¬ 
thetic rubber, and is probably rather beneficial. 
The only practical solvent known is benzine 
(not CgHc) and even this is not very good. 
Castor oil is a definite chemical compound, and 
is unstable when heated. 

Butyl Phthalate. This and similar organic 
liquids have been used occasionally with evident 
success for filling transducers. The specific 
acoustic impedances are usually close to that 
of sea water, and the viscosities are advanta¬ 
geously lower. 

Ethylene Glycol. This is usually used with 
water, as an antifreeze filling for domes. Ro¬ 
chelle salt is dissolved by ethylene glycol. 

Olive Oil; Peanut Oil. These liquids have been 
proposed as substitutes for castor oil to get 
lower viscosity. They have not been investigated 
in detail, but seem promising. There are many 
other such vegetable oils which also might prove 
useful. 

Silicones. The Dow-Corning Corporation has 
placed on the market a group of silicone fluids. 
The viscosities available extend over quite a 
range, and the temperature coefficient of vis¬ 
cosity is in orders of magnitude less than that 
of castor oil. By inference, the temperature co¬ 
efficient of specific acoustic impedance may also 
be markedly less. These fluids do not harm 
metals, rubbers, enamels, etc. Present cost for¬ 
bids general substitution for castor oil, but low 
melting point, high boiling point, low viscosity 
(range from which to choose) and low-temper¬ 
ature coefficients suggest that in specific appli¬ 
cations these silicones would be preferable to 
castor oil in transducers. 

Acetylated Castor' Oil. This liquid is one 
of the substitutes for castor oil proposed by 
Bell Telephone Laboratories. Although some 
advantage may accrue, UCDWR believes that 
this is overshadowed by the harmful effect on 
Cell-tite foam rubber. After prolonged immer¬ 
sion the foam rubber swells considerably, ap¬ 
pears to soak up the liquid, and certainly loses 
all of its isolation properties. It may be inferred 
that the liquid similarly affects solid neoprene 
and perhaps other rubbers. 


Petroleum Oils. These have been proposed as 
substitutes for castor oil, and one such trans¬ 
ducer was successfully built. They have the ad¬ 
vantage of low cost, ready availability, wide 
ranges of properties, and close manufacturing 
control. They attack natural rubber quickly, but 
some synthetics (neoprene) are not seriously 
affected. 

Desiccants 

Calcium Chloride. This common laboratory 
desiccant is frequently used in cloth sacks in 
transducers. Because of ions formed, it is not 
advisable to use it in the castor oil cavity. Its 
chief use is for drying air-filled cavities where 
electric connections are made. 

Silica Gel. This desiccant is very active and 
absorbs tremendous quantities of water; it may 
be used in cloth sacks in the oil cavity if desired 
and is generally preferable to calcium chloride. 

Phosphorous Pentoxide, etc. This and similar 
desiccants are not suitable for transducer use 
because of corrosive action. 

Isolation Material 

Corprene. This is a generic name for various 
compounds of natural, synthetic, and reclaimed 
rubbers (or sponges) with ground cork. Cor¬ 
prene is very commonly used to provide acoustic 
isolation in transducers; the specific acoustic 
impedance is undoubtedly low compared with 
water, but its value is unknown. There is reason 
to believe it to be frequency dependent and per¬ 
haps dependent on angle of incidence. At low 
frequencies it shows marked hysteresis which 
indicates that a sheet of it will absorb fair 
quantities of energy. The ideal isolator presents 
a purely reactive impedance, and this loss is 
undesirable. Corprene must be sealed over its 
entire surface (which is not easy) lest it soak 
up water, castor oil, etc., and change properties 
in time. Corprene is available in sheets, rolls, 
rods, and special moldings. It cuts easily with 
a knife, but accurate dimensions are best shaped 
on a disk or belt sander. Corprene cements 
easily to other surfaces, and Corprenes com¬ 
pounded from neoprene are not seriously dam¬ 
aged by petroleum derivatives. 

Foamgkis. This is a true foam (see foam rub¬ 
ber below) made of glass. The cells are reason- 




126 


COMPONENT PARTS 


ably uniform in size, perhaps % 2 - to %6-in. 
diameter. Foamglas, like glass, is impervious 
to nearly anything, and absorption is limited to 
what clings to its surface. Its specific acoustic 
impedance is quite low and appears to be largely 
reactive. It is not as good for isolation as Cor- 
prene or foam rubber. Foamglas comes in slabs 
of different sizes and can be cut to fairly accu¬ 
rate shapes with a knife. It can be machined 
in lathes, milling machines, etc., using a piece 
of scrap metal as a cutter. Foamglas will sup¬ 
port 150 psi when distributed over a few square 
inches, and will support the hydrostatic pres¬ 
sure of a couple hundred feet submersion in 
the ocean (exact depth variable and unknown). 
An outstanding property is the density: 0.17 g 
per cu cm. Foamglas is manufactured by Pitts¬ 
burgh-Coming Glass Company. 

Cell-tite Foam Neoprene. This material is 
available in sheets of varying thickness, the 
large surfaces being smooth. Internally, the 
sheet is composed of a mass of bubbles whose 
walls are neoprene. Since each bubble is sep¬ 
arated from its neighbors by a neoprene wall, 
this constitutes a true foam as distinguished 
from a sponge (such as a natural sponge) in 


which the cells are intercommunicating. A true 
foam will not soak up liquids except by diffusion 
through the cell walls. The specific acoustic im¬ 
pedance is quite low, probably lower than that 
of Corprene or Foamglas, and a thin layer 
seems to act as a remarkably good reflector 
with little loss. UCDWR tested foam rubber to 
250 psi and found that it continues to act pre¬ 
dominantly as air (pressure-volume product 
constant) ; the tests stopped here, and the upper 
limit is not known. Being neoprene, foam rub¬ 
ber is not harmed by any common substances 
in transducers except some organic solvents. 
There have been indications that under hydro¬ 
static pressure in castor oil foam rubber may 
lose its gas by diffusion and collapse. Attempts 
have been made to seal it with coatings but no 
results are yet available. So far there is no 
report of a transducer’s behavior changing be¬ 
cause of this collapse. It is readily cemented 
using bakelite cement, etc. UCDWR now uses 
foam rubber for nearly all isolation except 
where the rigidity of Foamglas or Corprene is 
required. Cell-tite foam neoprene is manufac¬ 
tured by Sponge Rubber Products Company, 
Derby, Connecticut. 




Chapter 4 

PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


By T. Finley Burke, Glen D. Camp, and Bourne G. Eaton 


" 1 GENERAL CONSIDERATIONS 

T he properties of the components of crystal 
transducers have been studied in Chapter 3, 
in so far as present experimental and theoret¬ 
ical knowledge permits. In this chapter, the 
properties of complete crystal transducers are 
studied, to determine to what extent these 
properties can be deduced from those of the 
components, and to try to come to a qualitative 
understanding of the discrepancies. 

The dependence of these considerations upon 
an admittedly incomplete theory should be em- 


Sound Field 



Figure 1. A crystal transducer as a “black 
box.” A lead from the backing plate is some¬ 
times brought out for experimental purposes, as 
indicated by the broken line. 

phasized. In the first place, it is important to 
notice that the study of acoustic vibrations is 
almost never associated with the direct meas¬ 
urement of acoustic quantities. For example, 
crystal transducers are calibrated against 
standards which have themselves been cali¬ 
brated by reciprocity measurements. At no 
point in this sequence is a mechanical quantity 
directly observed, all measurements being elec¬ 
trical and geometrical. Furthermore, granting 
the theoretical connections which yield acoustic 
quantities from electrical measurements, one 
would still be unable to interpret the empirical 
results in such a way as to motivate design 
studies, without an approximate theory. 

A complete crystal transducer presents itself 
to us as a “black box,” as shown in Figure 1, 
accessible only through its electric terminals 


and the sound field, the latter in the indirect 
way mentioned above. A calibration group will 
wish to emphasize this black-box viewpoint, to 
avoid subtly prejudicing their experimental 
procedures. 

For research and design purposes, one will 
try to understand the experimental results first 
as a consequence of the properties of the com¬ 
ponents and, if this is not possible, in terms of 
these components together with couplings be¬ 
tween them. It turns out that the principal fea¬ 
tures of a complete transducer are deducible 
from the properties of the components, but 
there are usually residual effects of varying 
magnitude in different units. Some of these 
residual effects are not understood at all, while 
others are understood only qualitatively. How¬ 
ever, as knowledge and design procedures have 
improved, the number of transducers having- 
small residual effects has steadily increased. 
Furthermore, the most satisfactory transducers 
are those in which these residual effects are 
negligible, both by empirical results and judged 
theoretically, the latter because every conceiv¬ 
able additional coupling has a harmful effect. 
Thus, much of the effort of a designer is di¬ 
rected toward rendering negligible those cou¬ 
plings which he has been able to identify. 


' ^ ^ Equivalent Circuit 

The existence of an equivalent circuit for a 
linear dissipative crystal transducer has been 
shown to be a consequence of the general theory 
(see Section 2.4), and has been verified within 
experimental error by reciprocity measure¬ 
ments. However, the values of the elements of 
this equivalent circuit and even the structure 
of the circuit are largely unknown. 

An equivalent circuit for a single crystal has 
been theoretically deduced in Chapter 3. This 
circuit includes the effects of finite lateral di¬ 
mensions and of normal and tangential imped- 


127 














128 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


ance loads on the radiating and lateral faces 
to the lowest order. For the present purpose, 
it is convenient to replace the direct capacitive 
coupling by an ideal transformer with a 1/1 
voltage ratio,to avoid confusing electric and 
mechanical grounds when more than one crystal 
is involved. The single-crystal circuit, with a 
transformer replacing the direct condenser cou¬ 
pling, is shown in Figure 2, which should be 
compared with Figure 5 of Section 3.2.1. 

The circuit of a complete transducer, neg¬ 
lecting all coupling between crystals, is obtained 



Figure 2. Equivalent circuit of rectangular 
crystal plate including perturbations to the 
Mason circuit, with ideal transformer instead of 
capacitive coupling. 

from that for a single crystal. Figure 2, by in¬ 
serting stray capacitances from each electrode 
to the backing plate (if conducting) and to the 
case, and then connecting the leads as they are 
connected in the transducer. This is shown in 
Figure 3 in which the dotted lines indicate a 
common type of electric connections, all crystals 
in parallel and the case grounded, the latter not 
being shown. This balanced drive is used in an 
effort to reduce the shunting effect of stray 
capacitances to the backing plate and case, 
which reduce the band width; it is found that 
these strays are very important and also, unless 
the utmost precautions are taken to keep every¬ 
thing symmetrical, the transducer will have an 
appreciable unbalance, still further increasing 
the shunting capacitance (see Section 4.5). The 
broken leads are to suggest the unknown me¬ 
chanical couplings here neglected. More than 
one is drawn from each element to emphasize 


“ In this volume, what might be called a com¬ 
pletely equivalent electric circuit is used; the trans¬ 
former converts volts to volts in the ratio 1 to 1, and 
the velocities and impedances in the mechanical arms 
are changed from mechanical units to electrical in the 
ratio 1 to 0 and 1 to Mason^ leaves the mechan¬ 

ical arms in mechanical units, but uses a transformer 
that converts volts^ to force in the ratio 1 to 0. Either 
convention leads to the same results. 


that these are the equivalent of a distributed 
impedance and that additional couplings might 
occur at any point. 


Possible Measurements 

The measurements which can be made on a 
complete transducer with present facilities are 
conveniently divided into two categories. 

1. Those in which only the electric terminals 
are accessible, the mechanical terminals being 
terminated in some manner which should be 
specified as sharply as possible, preferably by 
being coupled to an effectively infinite medium. 
These include impedances, both the two-terminal 
routine measurements and the multi-terminal 



Figure 3. Equivalent circuit of a complete 
transducer, neglecting the coupling between 
crystals. The broken leads from each lumped 
impedance are to suggest these neglected 
couplings. 


(usually 4) impedances measured for research 
purposes; crosstalk between two motors or be¬ 
tween two sections of the same motor; internal 
reverberation, either in steady state or for pulse 
excitation; and d-c resistance, of primary im¬ 
portance as a production test. 

2 . Simultaneous electrical and sound-field 
measurements. These include routine calibra¬ 
tion measurements of directivity patterns and 
various types of responses, and measurements 
of a similar character but made for research 
purposes. 

In addition to the above, it is believed that 
valuable information might be gained by 






























RADIATION THEORY 


129 


measurements made with the aid of probes at 
various points inside a transducer. Aside from 
the possibility of such measurements leading to 
a better understanding of the internal motion 
in transducers, probes might also be useful as 
a means of controlling the gain of a negative 
feedback amplifier; this might yield much 
greater band width, determined by the break¬ 
down voltage rather than the mechanical and 
electrical Q’s. This has never been tried, to our 
knowledge; there is one possible advantage over 
a filter-controlled amplifier, in that a crystal 
inside the transducer is the ideal equalizer for 
similar crystals (see Chapter 5). 

Considerable work has been done with probes 
at this laboratory (see Sections 3.4, 3.6, and 
4.7), but it is believed that further work might 
be very worth while. It might even be worth 
while to consider the possibility of building a 
stroboscopic optical interferometer to serve as 
a zero-impedance probe. 

Most measurements have been made in water, 
but it is believed that valuable information 
might be gained by making measurements in 
air, i.e., with the transducer diaphragm seeing 
zero impedance. Under these conditions, the 
crystals do not see zero impedance, since they 
work into a cavity usually filled with castor oil 
and bounded by steel, rubber, etc. The results 
would be difficult to interpret, but might never¬ 
theless be very useful. Two qualitative points 
are immediately clear: because of the smaller 
wavelength, directivity patterns are about five 
times as sharp in air as in water, assuming the 
same distribution of velocity over the external 
surface (this is easy to demonstrate qualita¬ 
tively) ; and, if the impedance of a transducer 
is found to be about the same in air as in water, 
then it must be dissipation controlled and hence 
be very inefficient. Window-coupled gas-filled 
transducers, of which this laboratory has built 
several, would be especially amenable to study 
in air, since removing the water termination 
would cause a readily calculable change in the 
impedance seen by the crystals. 

4 2 RADIATION THEORY 

The physical system composed of one or more 
transducers coupled to a real medium (e.g., the 


ocean plus a target) is so complicated that its 
study as a single problem is impractical. It is 
therefore very fortunate, on the one hand, that 
the exact nature of the source or receiver is not 
critical to the study of problems specific to the 
medium while, on the other hand, only the most 
primitive properties of the medium are relevant 
to the design of transducers. Knowing the char¬ 
acteristics of a transducer in an idealized me¬ 
dium and knowing the characteristics of the 
actual medium, the results to be obtained with 
the transducer in the actual medium, at least 
in principle, can be deduced.- For discussing the 
pure characteristics of transducers, therefore, 
the medium is idealized to an infinite fluid which 
is homogeneous, nonrefracting and nonabsorb¬ 
ing. It is therefore completely characterized by 
its density and phase velocity, o and c. Varia¬ 
tions of these quantities in the ocean with tem¬ 
perature, pressure, salinity, frequency, and any 
other variables, important as they are in the 
study of transmission, reverberation, etc., all 
have a negligible effect upon the performance 
of transducers. 


‘ ^ * The Sound Field 

The pressure produced by a given velocity 
distribution over a closed surface was shown, 
in Section 2.1.7, to be given by 

P(2) = -(^) J <iS.i-.(l)g,(l,2), (1) 

in which is the outward normal component 
of the velocity amplitude at each point on the 
closed surface and g,.{l,2) is the rigid Green’s 
function. 

This Green’s function, in an idealized fluid 
medium, depends upon the shape and size of the 
transducer surface and upon the wavelength, in 
addition to the points ri and r.y at which it is 
evaluated. In Section 2.1.7, it was stated with¬ 
out proof that, for a point r 2 , very far from the 
transducer and in the critical wavelength region 
in which most transducers lie, the best known 
approximation to the rigid Green’s function for 
a point ri on the surface and r^. very distant, is 

^,(1,2) = (14- cos a)^o(l,2), (2) 



130 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


god,2) 


exp I r2 — ri 

Ira - TiI 


(3) 


cos a 


(rz - ri) • Hi 

11*2 - ri 1 ’ 


(4) 


in which ni is the outward normal to the surface 
at the point ri, and a is the angle at ri between 
this normal and (r^ — 0). 

The origin may be taken at any fixed point, 
but for convenience it will always be assumed 
to be located at some point on or inside the sur¬ 
face of the transducer. Then 


n « rz, (5) 



Equation (6) gives the phase difference corre¬ 
sponding to the path difference |r2 — ri|. This 
has a principal term kr-. corresponding to the 
path difference from the reference point on or 
inside the transducer, a term kvi • Yo/to depend¬ 
ing upon To only through its direction, and 
higher-order terms which depend upon the 
magnitude of To. The radiation zone is defined 
as that part of the field beyond which the latter 
terms are negligible, and the present considera¬ 
tions are confined to this zone. The lowest-order 
term in equation (6), which depends upon the 
magnitude of Vn, is of the order of 


III 
Xrz ’ 


(7) 


It is convenient to separate the dependence on 
distance and direction into separate factors. 
One chooses a standard distance r,, which is 
usually 1 yd or 1 m. Also, a reference axis is 
chosen in the direction 0 ^. 1 ’^,ix referred to a 
specified frame of spherical coordinates; for 
convenience, this is usually chosen as the axis 
of maximum intensity, or a particular one of 
these if there are more than one. Then, if 2"^? 
is the pressure amplitude obtained from equa¬ 
tion (8) at one has 

p = exp ik(r. - r.) 2>PD(e,<l,), (9) 

= 1 . ( 10 ) 

The factor 2’- is inserted so that P itself will be 
the rms amplitude. 

The factor is called the amplitude- 

directivity pattern of the unit as a transmitter. 
Its angular variation is given by the integral in 
equation (8), 

D(d,4>) ~ 

JdSiVnd)d + cos a) exp^(11 ) 

and it is to be normalized by equation (10), 
The directivity patterns corresponding to equa¬ 
tion (11) for a variety of velocity distributions, 
together with other relevant matters, are dis¬ 
cussed in detail in Section 4,3, 

The intensity at any point on a sufficiently 
distant sphere is, from Section 2,1,7, directed 
radially outward and has magnitude 


and it turns out that, for moderately directional 
transducers, the error involved in dropping 
these higher-order terms in the phase will be 
negligible if To is about 20 wavelengths or 
greater. This indicates that in most cases cali¬ 
brations are done at an adequate distance, but 
there is no great margin to spare and some 
calibrations have been performed in the reor¬ 
ganization zone. 

Assuming that the phase error roughly given 
by equation (7) is negligible, the pressure is 



In equation (12), \P\-/pC is the total response, 
for the given surface velocity, and \D{6,(i>) |- is 
the intensity-directivity pattern. The total re¬ 
sponse is variously expressed as 

= TjP, etc,, (13) 
pc 

in which T^, T^, etc,, are the transmitter re¬ 
sponses per volt across specified terminals or 
per ampere in these terminals, etc. These spe¬ 
cific responses, and the intensity-directivity 
pattern, are usually expressed in db relative to 
the appropriate reference level. 













RADIATION THEORY 


131 


The total power radiated is the flux of the 
intensity across a large sphere, 

[DIO- e,<t> 

Total power = A-n-r'i- —^ |i)(0,0) |“ , (14) 

pc 

\D{e,<t,)\"’* = (^) J iD(e,4') 1’. (15) 

in which dco is an element of solid angle. 

The average of the intensity pattern, equa¬ 
tion (15), is called the directivity factor. The 
quantity obtained by converting this to decibels, 
- 6,<t> 

10 log \D{d,(f>)\^ , is called the directivity 

index ; the directivity factor and index of a per¬ 
fect spherical radiator are 1 and 0 db, respec¬ 
tively, and are less for all other radiators. A 
rough estimate of the directivity factor of a 
transducer, suitable for order of magnitude 
considerations, is obtained by taking the ratio 
of the solid angle in which the pattern is above 
—3 db to that of all space, 4jt. For example, sup¬ 
pose a pattern has one fairly symmetric main 
lobe with a 3-db half-breadth of 6*^ radians; then 
the directivity factor is roughly 

=sirf(|). (16) 

If is a fraction of a radian, as it will be in 
many cases, then this is roughly (6^/2)-; thus 
the directivity factor of a single lobe unit is of 
the order of one quarter of the square of the 
(radian) half-breadth. 


Radiation Impedance 

The calculation of the radiation impedance 
corresponding to a given surface-velocity dis¬ 
tribution is one of the most important and, at 
the same time, one of the weakest parts of 
present theory. This complicated matter is dis¬ 
cussed in detail in Section 2.1.7, where our ig¬ 
norance of the rigid Green’s function for vari¬ 
ously shaped bodies, in the critical wavelength 
region, is emphasized. 

Present knowledge can be summarized 
roughly as follows. 

1. If the radius of curvature of the surface 
is small compared with the wavelength, and if 


the normal velocity is roughly constant in a 
region with dimensions of the order of 1 wave¬ 
length, then the radiation reactance is negli¬ 
gible and the radiation resistance is approxi¬ 
mately QC. 

2. If the surface is covered by pistons (crys¬ 
tals) whose distance from center to center is a 
little less than a wavelength (about 0.8 will do; 
see Section 4.2.4), then the wave front is “well 
supported,” the reactance is again negligible, 
and the resistance is qc times the ratio of the 
active to the total area. 

3. If the wave front is not “well supported,” 
or if there are appreciable variations of the 
normal velocity in a distance of the order of a 
wavelength, then the reactance will have a value 
which is an appreciable fraction of qc, while 
the resistance will fall from its full qc value. 

These semiquantitative results are illustrated 
by the impedance of a circular piston in an 
infinite baffle,^ but it should be emphasized 
that the results for a piston in a baffle are not 
quantitatively applicable to other surfaces. 

The physical basis of these results is that a 
small piston will have particle-velocity stream¬ 
lines in front of it which diverge, allowing the 
fluid to escape laterally; however, if the smallest 
dimension is increased to a wavelength or so, 
the escape will become small and the piston will 
begin to behave like an infinite plane. 

There is one other attack that gives some 
information about the resistive part of the radi¬ 
ation impedance, and this may be quite accurate 
in the case of a normal velocity which is fairly 
constant over the vibrating diaphragm and zero 
elsewhere. This arises from the fact that we 
have a better approximation to the surface- 
distant than to the surface-surface Green’s 
function. If the normal-velocity amplitude is 
constant over a surface, then the power radiated 
is r^^A\v\-/2 (the factor 1/2 enters because v is 
the peak, rather than the rms, velocity ampli¬ 
tude), in which r is the average specific resist¬ 
ance seen by the piston. Combining this with 
equation (14) for the total power radiated, one 
has 


/ 47rr^ 1 P 1 2 \ —-- 

To use this approximate formula, one calculates 










132 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


P from equation (8) ; the constant velocity v 
will be a factor of P, and hence the 1^1“ in the 
denominator of equation (17) will cancel out; 
likewise, rf will drop out. Furthermore, for 
most units the calculation of P, which amounts 
to calculating p(2) on the axis, is very much 
simplified since, for example, a fiat unit will 
have a zero phase shift A^ri • rs/ro. The calculated 
directivity factor should be compared with cali¬ 
bration data, if possible. 

It is extremely unfortunate that we are in 
such a weak position with regard to a quantita¬ 
tive treatment of the radiation impedance prob¬ 
lem, because the resistive part of this imped¬ 
ance not only determines the power output but 
also the band width of a transducer. However, it 
is important to clearly recognize this weakness. 

On the one hand, we have the theoretical dif¬ 
ficulties mentioned above while, on the other, 
the low electromechanical-coupling coefficients 
of 45° Y-cut Rochelle salt [RS] and 45° Z-cut 
ammonium dihydrogen phosphate [ADP] make 
it difficult to deduce the impedance of the me¬ 
chanical arm from measurements at the electric 
terminals. 

In Section 9.5 it will be pointed out that, since 
the electromechanical-coupling coefficient of 
45° X-cut RS at some temperatures is much 
greater than for either of the above crystals, 
the measurement of radiation impedance could 
be obtained from experimental transducers 
using X-cut motors, where the mechanical arm 
is not as badly obscured by the electric con¬ 
denser, so that present impedance bridges can 
resolve it. Since the radiation impedance de¬ 
pends only upon the wavelength in water and 
the geometry of the transducer, the results so 
obtained would be applicable to any type of 
crystal. 

The precautions of temperature control, etc., 
necessary to make this method successful, are 
discussed in Chapter 9. 


Reciprocity 

In this section, the general reciprocity theo¬ 
rem of Section 2.4 is put into a form suitable 
for practical application to crystal transducers. 
The reciprocity relations are the basis of the 


absolute calibrations of electroacoustic trans¬ 
ducers, that is, their calibration in terms of 
electrical and length measurements only. It is 
therefore important to understand the assump¬ 
tions upon which they are based. 

The first assumption is passivity, no sources 
of energy. This is satisfied in all units except 
those containing a preamplifier. The second is 
linearity, both for transducers and medium. We 
have every reason to believe that this assump¬ 
tion is satisfied far beyond any practical ac¬ 
curacy in the medium and in ADP and Y-cut RS 
units, the first two probably better than the last 
because fringing flux involves the nonlinear 
relation between electric displacement and field 
in the X-direction of RS. It is definitely not 
satisfied for X-cut RS, and hence such units 
should be calibrated by comparison with linear 
units that have been calibrated by the absolute 
method. In any event, the two assumptions can 
be tested by electrical measurements alone. 

These two assumptions can be expressed in 
the form 

Pf = TI, (18) 

e = Rp/. (19) 

The first asserts that the free-field pressure in 
the medium is proportional to the current in¬ 
serted into a unit acting as a transmitter; the 
second that the open-circuit voltage produced 
be a free-field pressure is proportional to that 
pressure. The quantities T and R depend on 
distance and direction, properties of the me¬ 
dium and transducer, and frequency, but are in¬ 
dependent of level. 

The third and last assumption is that any two 
units, fixed in position in the medium, will be¬ 
have like a four-terminal electric network. Thus 
if I inserted into No. 1 produces e in No. 2, it 
is assumed that the same I inserted into No. 2 
will produce the same e in No. 1, 

6 = RiTiI = RiToI. ( 20 ) 

This is a property of a broad class of systems, 
called linear, passive, and bilateral. It is shown 
in Section 2.4 that a quite general class of 
linear-dissipative piezoelectric systems are cir- 
cuit-like. 




RADIATION THEORY 


133 


From equation (20) we conclude that 

fa’ 

This shows that the directivity pattern of a 
unit, satisfying the above assumptions, is the 
same either as a transmitter or a receiver. And 
more than this, that the ratio of the response as 
a receiver to that as a transmitter (both in the 
same direction) is a number which is not only 
independent of orientation, but does not depend 
upon the properties of the unit. It can therefore 
depend only upon separation, properties of the 
medium, and frequency. This quantity, com¬ 
monly called J, can therefore be evaluated by 
considering an idealized transducer. 

A spherical unit, radius a, is assumed to be 
so constrained that it can have only uniform 


K(z) = 


(26) 


/p\ _ ipchojka) _ 4>^Zm 
\v)a K{ka) A-rra-’ 


(27) 


in which corresponds to but is specific and 
measured in mechanical units, and the prime 
means differentiation with respect to the whole 
argument ka. Inserting the value of v from 
equations (22) and (24) into equation (25), 
we obtain 


p = TI, ■ (28) 

r = ,w-.(f+ (29) 

The equivalent circuit as a receiver is shown 
in Figure 5, in which it is important to notice 



Observe Im=(l>v 
Or Corresponding p 


Figure 4. Equivalent circuit of a transducer 
used as a transmitter. 


o-I 1^-o 

Observe e [S Insert E * (p-i4ira2p(a) 

O I o 

Figure 5. Equivalent circuit of a transducer 
used as a receiver. 


radial motion. Its equivalent circuit as a trans¬ 
mitter is shown in Figure 4, the assumption of 
this circuit corresponding to the assumptions 
discussed above. Capital Z, with various sub¬ 
scripts, is here used as the actual or equivalent 
electric impedance, 

The current I produces a current in the 
impedance which represents the medium, 
and this corresponds to a mechanical velocity v, 

Ln = (f + (22) 

Z = Z 2 + Z., (23) 

(24) 

in which is the electromechanical-coupling 
coefficient (whose value need not concern us be¬ 
cause it drops out of our final result for J ), and 
Z is the impedance seen looking into the me¬ 
chanical terminals with Z„^ removed and the 
electric terminals open. 

The above velocity is purely radial, and there¬ 
fore produces a pressure 

ipcvh(i{kr) 

^ ~ hl,(ka) • 


that here p{a) is the actual pressure, freefield 
plus reflected wave, at the spherical surface. 

We must now relate the actual pressure to the 
free field. Let a plane wave Pf = Po exp (ikz), 
traveling in the positive z direction, fall on the 
unit acting as a receiver. This plane wave may 
be regarded as the superposition of incoming 
and outgoing spherical waves (origin at center 
of receiver). These will have angular depend¬ 
ence according to the Legendre polynomials 
P„ (cos 6) and since the unit is assumed to have 
infinite impedance for all of these, except n = 0, 
(uniform radial motion), they will be reflected 
as if from a rigid sphere and may therefore be 
ignored. We therefore need consider only the 
spherically symmetric term, which can be ob¬ 
tained by averaging the pressure over any 
sphere of radius r, 

Pj J' ®xp (ikr cos 0) d9. 

= Pojo (hr), 

Jo (kr) = (30) 

To this free-field pressure must be added a 


(25) 















134 


PROPERTIES OE ASSEMBLED CRYSTAL TRANSDUCERS 


wave which makes the velocity on the surface 
correct, and which must vary like Kikr) since 
it is outgoing, 

P = Po [joikr) + fiho(kr)], (31) 

— d)“Z 

p(a) = poUo + i^ho) = -zv = - 4 ^, (32) 

(^) ^^0 ty'o + ^^h'o] = io^pv, (33) 

in which 2 : corresponds to Z but is specific and 
in mechanical units, and the argument of jo, ho, 
etc., is understood to be ka when not written. 

Dividing equation (33) by equation (32), we 
eliminate v and po and thus find p; putting this 
back into equation (32), the desired relation 
between p (a) and Pf(a) is established, so that 
we can find v in terms of Po, 

This velocity, produced by po, is equivalent to a 
current (fiv, and the drop across Z^ is therefore 

e = 4>vZc = Rpo, (35) 

R = ^ J {Kz + ipcho)-K (36) 

We thus find for the ratio J, 


T =^= 

T k'"a- (z + Zm) pcho (kr) 



and equation (19) becomes 

e = JTpf. (39) 

The absolute evaluation of T, and therefore 
R, is now seen to be achieved by running three 
units against one another, at least one being 
used both as a transmitter and a receiver, ac¬ 
cording to either of the two schemes: 

Transmitter 12 3 12 1 

(40) 

Receiver 2 3 1 2 3 3 

This gives the three products TxT,, and 

T^Tx in terms of electrical (and distance) meas¬ 
urements only. 

In conclusion, we notice that a reciprocity 


measurement by itself is not an absolute cali¬ 
bration unless the units and the medium have 
been shown to satisfy the basic assumptions at 
the actual operating levels. For example, if one 
of the transmitters causes cavitation, the whole 
measurement is invalidated. 

The above discussion neglects the effect on 
the transmitter of the wave scattered by the re¬ 
ceiver. This may be important since calibrations 
are usually done at fairly short range. The 
usual test is to see if the inverse square law of 
intensity is satisfied, and this seems to be en¬ 
tirely adequate. 

We observe from equation (38) that 

iT|= = (41) 

Thus if a correction of 6 db per octave is added 
to the receiver response (open-circuit voltage 
per dyne per sq cm, free field), the resultant 
curve should coincide with the transmitter re¬ 
sponse (free-held dynes per sq cm per amp) if 
it is displaced vertically. Furthermore, the nec¬ 
essary vertical displacement of the corrected 
receiver response should be given by 20 
log (p/2r). This relation is useful as a check on 
the consistency of calibration data. One usually 
hnds the above relation to hold within a db or 
so over a wide frequency range; this not only 
gives conhdence in the accuracy of the calibra¬ 
tion data, but also in the validity of the reci¬ 
procity assumptions. 

A similar relation exists between the 
short-circuit receiver response (amperes per 
dyne/per sq cm free held, when receiver termi¬ 
nals are short-circuited), and the constant- 
voltage transmitter response (free-held dynes 
per sq cm per v). It is unfortunate that the 
short-circuit receiver response is not used more 
widely as a calibration method, since it elimi¬ 
nates errors arising from cable and stray ca¬ 
pacitances. 


Cylindrical Transducers 

Transducers whose active faces form arcs of 
circular cylinders are of such practical impor¬ 
tance as to be worthy of special consideration. 
The considerations developed in this section 






RADIATION THEORY 


135 


will also have application to other curved-face 
units. 

It is advantageous to first restrict ourselves 
to a complete cylindrical arc. Neglecting the 
slight deviation of the flat radiating faces of the 
crystals from a cylindrical surface, let N equal 
pistons of height 2h be uniformly arrayed so as 
to form strips on a circular cylinder of radius a, 
the centers being at 2k7i/N radians (n = 1 to 
N) and the angular width being 2a. We want to 
know how large N must be to avoid a serious 
scalloped appearance of the pattern, and how 
to calculate the impedance and response. 

Directivity Patterns 

The pressure at r 2 , dropping all factors inde¬ 
pendent of direction, is given by 

P dSiu{l) exp ^ -i^ri • ~ J * 

in which the large R’s are cylindrical, the small 
?'’s spherical, radial coordinates. It is important 
to include the obliquity factor, last factor in 
the integrand, in treating cylindrical units to 
prevent spurious contributions through the 
cylinder. In this factor, ^ is the impedance of 
the medium at each point on the surface, in 
oc units. In all that follows, we assume i;(l) to 
be a constant on all pistons and zero between 
them. 

The integration with respect to Zi gives a 
factor 

sin {kh sin 

{kh sin \^ 2 ) ’ ^ ’ 

in which sin = Zo/r^ is the angle from the 
equatorial plane. This factor is identical with 
that for a line source. 

The obliquity factor can be generated by a 
differential operator, and the remaining inte¬ 
gral therefore becomes 



exp [ - ika cos (^2 — (pi) cos \p 2 ]. (44) 

The above integrand can be expanded in a 
Fourier series^ according to 

exp iz sin (p = ’Jm(z) exp im<p. (45) 


The integration extends between <pi = 2iin/N 
— a and 2nn/N -f « for the nt\\ piston and 
this must then be summed over all pistons. The 
result is that only those m’s which are integer 
multiples of N contribute. These give a factor, 
sin \iNci/\iNa. 

Dropping further unessential factors, the 
pressure pattern is 

/ sin kh sin ^Pi \( . i 
^ \ khs,m 4^2 dkaj' 

X “ f) ’^2). 

(46) 

This has a term independent of (p 2 , together with 
terms which give N, 2N, etc., maxima as <p 2 
ranges from 0 to 27r, The amplitude of these 
scalloped petals is determined by the factors 
(1 + i^d/dka) • (sin fxNa/iJiNa)J^x{ka cos i/'o). 
If Na = X, the cylinder is completely filled up, 
all sine factors (except for m = 0 ) vanish and 
the pattern is uniform. However, it is usually 
not possible to fill the arc well enough to get 
much help from the sine factors, and we must 
therefore look at the Bessel factors. Dropping 
all but /X = —1, 0, and 1, and confining our¬ 
selves to the equatorial plane where the scalloped 
effect is worst, we have 

p ^ (Jo i J()) + 2( — i)'' (e/^v + i Jy) - 
(smNa\ 

In most cases of practical interest, the imped¬ 
ance factor 'C, will be very close to 1, and to sup¬ 
press the scallops we must therefore take N 
large enough that 


e/,v + iJx I .. 
Jo + iJol 


(48) 


The Bessel functions, considered as functions 
of their parameter N for fixed argument ka, 
oscillate out to N — ka and then fall away expo- 
nentially.'^ This transition from oscillatory to 
exponential behavior occurs with a change of 
only 1 in N and we therefore need only choose 
N a little larger than ka to make the pattern 
uniform. The amplitude of the scallops in any 











136 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


particular case is readily calculated from equa¬ 
tion (47). 

Impedance 

To calculate the impedance of units which 
are at least a few wavelengths high (the prac¬ 
tical case, since any vertical directivity requires 
this), we can consider the unit to be infinitely 
long without serious error. A normal-mode ex¬ 
pansion is then taken for the pressure, since 
we need to know it and its radial gradient at 
the face, 

( ipcvNa \ ^ / sin pNa \ 

ihcuia) = 

It is readily verified that equation (50) gives 
velocity v on each crystal strip and zero between 
the strips. 

The average pressure acting on a crystal 
strip is 

P = J d4>p{a), (51) 

strip 

and this just inserts another factor (sin 
[iNa/\iNa) into equation (49). Now if N has 
been taken large enough to make the pattern 
fairly uniform, only the term p = 0 will con¬ 
tribute appreciably to equation (51), because 
the factor falls rapidly as N exceeds 

/m, and we have 



This is just {2Na/2n) times the pressure pro¬ 
duced by a complete arc vibrating radially and 
we have the physically plausible result that the 
impedance seen by each crystal is merely that 
for a completely filled cylinder multiplied by the 
ratio of the driven area to the total area. 

Incomplete Arc 

The foregoing refers to a uniform angular 
distribution of pistons, and we need to know 


how our conclusions will be altered if there are 
only N' pistons with the same center-to-center 
angle 27i/N. 

In this case, values of m other than integer 
multiples of N make contributions, and equa¬ 
tion (46) is replaced by 


( 


sin kh sin \l /2 


)( 


1 + IM P 

^ dka I 


kh sin \l /2 
S ^,n(ka cos h) exp im ( 02 - 


(DP 


'sin TmN'\ 
N I 


exp 


[ 


-Trim (AT' + 1) 
N 


} 


(53) 


The physics of this complicated result can be 
understood well enough for all practical pur¬ 
poses as follows: we restrict our considerations 
to the equatorial plane, and assume that ka is 
large enough to make u close to 1. Furthermore, 
we suspect that just as in the case of the com¬ 
plete arc, we will need to take N a little larger 
than ka to prevent the pattern showing a maxi¬ 
mum for each piston. 

The factors J,„ -f ?'/„/, obtained by allowing 
the obliquity operator to act on will be well 
represented by their asymptotic expansions out 
to \m\ — m', the integer nearest ka, and will 
then fall so sharply that we can break the sum 
there. Since m' is less than N and a is less than 
%/N, the factor (sin m.a) / (mn/N) is approxi¬ 
mately (Na/jx), independent of 7n. Dropping all 
factors independent of m and </) 2 , the equatorial 
pattern is therefore approximately 


p ~ XI exp im4>o 
exp 



(54) 


and it is easy to verify that this is exactly the 
first 2 m' -j- 1 terms of the Fourier expansion of 
a function which is constant in front of the 
active face (including dead strips) and zero 
elsewhere. Now the effect of the high terms 
absent in this series is well-known; they are 
necessary to give sharp corners and complete 
constancy. We therefore conclude that if ka is 
large (at least 2 or 3, which is always neces¬ 
sary anyway to get appreciable vertical direc¬ 
tivity in a unit of practical size), and if A is a 




















RADIATION THEORY 


137 


little greater than ka, then the pattern corre¬ 
sponding to uniform velocity over an incom¬ 
plete arc is simply a cylindrical wedge in front 
of that arc, with small deviations from con¬ 
stancy, the corners rounded, and some small 
spill-over at the edges of the wedge. The angle 
at which the pattern is down, say, 3 db from the 
center value, could be calculated by numerical 
evaluation of equation (53). However, the ve¬ 
locity is never quite constant over the active 
face, either in magnitude or phase; the effect of 
obstacles (walls of the case, etc.) have not been 
considered; etc. For these and other reasons, 
this computation is not worth making; it is 
much better to use the foregoing theoretical 
treatment as a semiquantitative guide to de¬ 
sign, build a unit, and see how it works. 

We are now in a position to make some 
further semiquantitative generalizations from 
the foregoing results, together with similar re¬ 
sults for a spherical transducer. The condition 
that N must be a little larger than ka can be 
written 


and in this form it has a very simple physical 
interpretation which with an obvious slight 
alteration is applicable to noncylmdrical units. 
The left member is just the arc distance be¬ 
tween centers of adjacent pistons and equation 
(55) therefore asserts that to radiate a good 
cylindrical wave, that wave must be supported 
at points less than a wavelength apart. We 
could have written this down at the outset, but 
the analysis has revealed one rather surprising 
quantitative result: the suddenness with which 
good cylindrical waves set in as N passes the 
value ka. This is a consequence of the exponen¬ 
tial decay of /,„(fca) after m passes ka, and 
would never have been predicted from qualita¬ 
tive physical arguments alone; one would more 
likely have expected that the wave would need 
support at many points within a wavelength 
whereas, actually, if it is supported an average 
of 1.2 times in every wavelength, the pattern 
is excellent. As a practical example of this, we 
need only examine the patterns of complete arc 


units actually built and see how suddenly a 
breakup occurs from quite a smooth pattern to 
a scallop with, at first, N petals as the frequency 
passes through the critical value. Once this 
breakup occurs, the pattern is controlled by Jy 
until the frequency is approximately doubled, 
when J 2 X suddenly becomes important. For in¬ 
complete arcs, the situation is a little more com¬ 
plicated; up to ka a little less than N, the pat¬ 
tern is the rounded wedge previously discussed, 
but as ka exceeds N everything happens at 
once. It is of no importance to attempt to calcu¬ 
late these complicated effects; rather, we must 
merely arrange our designs so as to avoid them 
and this is very fortunately a much simpler 
matter. Calculations for flat and spherical units 
lead to the same conclusions as to the effect of 
spacing the active elements and we can there¬ 
fore state a general spacing rule in a form ap¬ 
plicable to any transducer, flat or curved: to 
generate a ivave surface that ivill not contain a 
resolution of the active elements, the largest 
cente7'-to-ce7iter distance betwee^i active ele- 
meyits must he slightly less than a wavelength 
(not more than 80 per cent). 

It may at first glance seem curious that the 
size of the active elements {2aa in the cylin¬ 
drical case) does not enter into this rule, i.e., 
that a number of points of negligible size com¬ 
pared to the wavelength, but spaced with a lat¬ 
tice constant of not more than 80 per cent of 
a wavelength, will give a smooth pattern. The 
essential point is that interference between the 
subelements of an element does not become im¬ 
portant until the dimensions of the element are 
nearly a wavelength, and then only at angles 
considerably removed from the normal. If we 
keep the center-to-center distance between ele¬ 
ments fixed at something less than a wavelength 
and then allow the elements to grow until they 
fill the surface completely, the only appreciable 
effect is therefore a gradual increase in inten¬ 
sity. This result is merely another aspect of the 
well-known principle in optics that two objects 
separated by less than a wavelength are not 
well resolved. While the size of sufficiently close 
elements has a negligible influence on the pat¬ 
tern, the change in intensity with size changes 
the impedance seen by each element, as dis¬ 
cussed below. 




138 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


Impedance of Incomplete Arcs and 
Other Surfaces 

In order to find the power radiated, we need 
to know the impedance seen by the active ele¬ 
ments. These are assumed to be sufficiently 
rigid that they move as a unit, so that we need 
to know the average pressure over their sur¬ 
faces, as a function of their velocity. 

If a single crystal with radiating dimensions 
small compared to the wavelength pushes into 
a fluid, the streamlines will tend to slip around 
to the side and the crystal will see a low radia¬ 
tion resistance. If we support this crystal by 
neighbors moving in phase, this divergence of 
the streamlines will be reduced. Now if the dis¬ 
tribution of the active elements satisfies the 
spacing rule for obtaining a good pattern, then 
the wave surface a w^avelength or so in front 
of the face will have lost most of the spottiness 
arising from the individual elements, and this 
rapid transition zone therefore functions essen¬ 
tially as a mechanical transformer. This means 
that equation (53) is essentially a local rela¬ 
tion and we therefore expect the analogous re¬ 
lation to hold on any surface satisfying the 
spacing rule. 


P 

V 


/ ^ \ iHo{ka) 
'^\A,l Hiika) ~ 



(56) 


in which and A, are the active and total 
areas. 


Response 

We are now able to calculate the response, in 
terms of the response of a related flat unit, of 
any unit that satisfies the spacing condition, 
which is of course the only type of unit in 
which there is much practical interest since one 
of the specifications is bound to be a pattern 
that does not resolve the active elements. 

Let P be the total power radiated, h the axial 
intensity, and D(6,(}>) the (intensity) direct¬ 
ivity pattern, one in the axial direction, so that 
I = IqD. Then 

P = J D sin eddd 4 > = iirf-hD, (57) 

sphere 

in which D is the average value of D over a 
sphere of sufficiently large radius r surround¬ 


ing the unit, and is called the directivity factor. 
The axial direction is always chosen as the di¬ 
rection of maximum intensity, so that D is 
always less than 1 , except for a spherical radia¬ 
tor, for which it is 1 . 

Now let us build a flat unit, using identical 
crystals again arranged so as to satisfy the 
spacing conditions. For example, corresponding 
to a cylindrical unit of total angle 27iN'/N, 
radius a, and height 2h, we build a flat rec¬ 
tangular unit of dimensions 2h by 2jtaN'/N. 
The power radiated, etc., from this unit is des¬ 
ignated by a prime. P' will differ from P only 
because of the change in impedance seen by the 
crystals, caused by a small change in the ratio 
of active to total area. The ratio P/P', different 
from 1 because of this impedance change, will 
depend upon the drive (e.g., constant current, 
constant voltage, out of a given amplifier), the 
matching network used, and the mechanical 
termination of the crystals (backing plate, in¬ 
ertia drive, etc.). In any event, this ratio is 
readily calculable from the equivalent circuit of 
the transducer and the two impedances; usually, 
P P' will correspond only to a decibel or two. 
The axial intensity /„ will differ from Iq because 
P' differs from P, but more important, because 
the total power radiated goes into a smaller 
solid angle, i.e., because D' is less than D. 

We can therefore compare the axial intensity 
for a curved unit with that of the corresponding 
flat unit: 

P = ( ) 4^r-I!,D' = i,r-hD, (58) 

10 log /„ = 10 log - 10 log I# J 

+ 10 log ( ^ (59) 

For a circular arc of total angle (f>, the change 
arising from the greater angle into which the 
energy is radiated is approximately 

-10 log ^ ^ j = -10 log cf>ka. (60) 

This can cause a considerable reduction in in¬ 
tensity compared to the corresponding flat unit. 
For example, our CS-type units operate at ka 
^ 10, (f> = 2 jt , and equation (60) gives about 
—18 db; our CQ units operate at a much higher 





DIRECTIVITIES 


139 


ka, about 30, but their arc is only 90°, so that 
equation (60) gives about —17 db for them. 

Since the open-circuit response of a unit as a 
receiver is simply related to its constant-cur¬ 
rent response,'^ we see that the same ratio of 
directivity factors will enter in the receiver re¬ 
sponse of a curved-face unit relative to that of 
a flat unit. 

However, in this case the physical interpre¬ 
tation is as follows: a plane wave falling on the 
unit excites the crystals lying in the first Fres¬ 
nel zone approximately in one phase, whereas 
the remaining crystals are excited in all phases 
about equally so that their resultant is small. 
For a cylindrical unit, the arc of the first Fres¬ 
nel zone is approximately so that a frac¬ 
tion («a) are excited in the zone. Since the 

effect of the others is negligible, we can regard 
them as shunts across those in the first zone 
and the voltage across any one is therefore only 
the above fraction of what it would be if they 
were all excited in phase. On a decibel basis, 
therefore, we expect the response to be changed 
on this account by 

-10 log -10 1og27r^a, (61) 

CLK 

which is the same as equation (60) for a com¬ 
plete arc. 


DIRECTIVITIES 


at the points of the sources, so that the wave 
motion at any point in space ahead of the wave 
front can be calculated from surface integra¬ 
tions over these assumed sources. 


u{x,y,z,t) 


1 r r r d ja- (ri - coai 

47rJ J 


d ilc (ri - ct)/r 


'] 


ds, 


« (62) 
(v is the velocity potential of the given wave at 
surface s) where dv/d7i is the normal derivative 
of the velocity potential at the surfaces s, n is 
the distance between the surface element ds and 
point {x,y,z). The first and second terms in the 
integral represent the simple and double 
sources respectively. 

There are three cases where the above inte¬ 
gral can be handled rather simply, i.e., waves 
that are plane, cylindrical, and spherical. The 
velocity potential of these can be written re¬ 
spectively : 


Vp = 

Vs = 

Tic = Yo{kr)e~^'"", 

YoiKr) =— log {^kr)Joikr)f 

TT 

As the spherical wave represents a point source 
and gives a nondirective field, it is trivial for 
use in transducer design. The plane and cylin¬ 
drical cases, however, are useful in that most 
transducers have either plane or cylindrical 
radiating faces. 


Theoretical Discussion 


Specific Examples 


The calculations of the directivities of trans¬ 
ducers leads at once into the general problem 
of the determination of a wave motion through¬ 
out an unlimited medium caused by an initial 
disturbance in a small part of the medium. This 
problem was first solved in general form for 
periodic waves by Helmholtz.*’* His solution, in 
effect, assumes that any wave front can be re¬ 
placed by a distribution of simple and double 
sources whose intensities are functions of the 
velocity potential of the wave being considered 

b See Section 4.2.3 on reciprocity. One subtracts 10 
log (10^ o/y2r)2 db from the transmitter response, db 
above 1 bar/amp, to get the open-circuit response, 
db above 1 v/bar. In this formula, all quantities are in 
cgs units. 


Plane Radiators 

As a large class of transducers have plane 
radiating surfaces, the case of the plane pro¬ 
gressive wave is of great importance. The ve¬ 
locity potential of this wave can be written 
u = in which case the general for¬ 

mula can be written in 

u(x,y,z,k) 

1 + cos ^ cos 'I' J (is — 0 , (63) 

where Vi is the distance between the area ele- 

c Yoikr) is Weber’s function.'^ Any Bessel’s function 
of order zero that has singularity at r = 0 can be used. 







140 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


ment ds and the point (x,y,z) and 'P is the angle 
between the x axis and Vi at the element ds. 
Two cases arise now, the one for points at large 
distance from the plane x — 0, and the other 
for points in the neighborhood of a; = 0. In the 
first case k » 1/ri, and the term l/vi cos 
can be dropped. Also as 


^ — jUf,mx-ct) 

dx ~ 


[li] = 

the equation becomes: 

u{x,y,z,t) = ^ J '^)ds. 


(64) 

The effect of the assumed sheet of double 
sources in the case of a plane wave thus adds the 
“obliquity” term cos 

In the second case, where ri is not very much 
larger than a wavelength, the third term must 
be retained. This problem was first solved rigor¬ 
ously by Sommerfeld® for the case of diffraction 
of a plane wave around a straight edge. His so¬ 
lution, in effect, describes the physical process 
as an interference between a plane progressive 
wave sharply cut off at the geometric shadow 
and a cylindrical wave generated at the edge of 
the opaque boundary. The diffraction of a slit 
would then be described as an interference be¬ 
tween three wave systems, two cylindrical with 
origins at the slit edges, and one plane, whose 
boundaries are the geometric shadows of the 
slits. The wave field immediately in front of the 
slit would then have many maxima and minima 
depending on the ratio of the slit width to the 
wavelength, while at great distances the field 
would display a central maximum with regular 
lines of maxima and minima on either side. 
Proceeding from the slit on perpendicular axis, 
then, there would be alternate maxima and 
minima out to about 10 slit widths, from which 
point on out the field would decrease steadily as 
the inverse square of the distance. The wave 
field close to the slit has its optical analogy in 
the Fresnel diffraction, and the field at great 
distance has its optical analogy in the Fraun- 
hoffer diffractions.The case of the infinite slit 


Excellent pictures of the complete sound field are 
given.o 


is not, except for a few cases, directly applicable 
to diffraction fields in high-frequency sound be¬ 
cause no transducers are ever built with infinite 
lengths and many do not resemble slits, and 
even to resemble slits or holes, the unit would 
have to be operated in an infinite baffle; but 
the general picture is the same for all sources 
that in size are comparable to the wavelengths 
they generate. The sound field or any shaped- 
plane radiator should, close to the surface, ex¬ 
hibit many maxima and minima distributed in 
space both along the axis perpendicular to the 
surface and in planes parallel to the surface, 
and should, at large distances, exhibit a central 
maxima with side lobes of decreasing ampli¬ 
tude with distance from the central axis. The 
field at great distances is of course important in 
sound signaling, and the field close in is impor¬ 
tant in the coupling between two or more trans¬ 
ducers that must be operated close together. 

Most of the plane radiators in use are 
bounded by squares or circles, and the interest 
is only in the distant part of their sound fields. 
Under these conditions, the directivities are 
easily calculated. The case of the circular 
radiator was first solved by Rayleigh^® who, 
neglecting the obliquity factor cos Tb evaluated 
the integral 



where s is a circular area of radius r, \_dii/dx^ is 
constant over this area, and r is the distance 
from element ds to point {x,y,z ). 

As the pressure p is given by p = —o du/dt, 
the directivity formula for the pressure can be 
written 


P{x, y, z) 



Integrating over the disk of radius r the pres¬ 
sure at point p is 


r n 27 rR'^ 

Ji{x) 

c 

o 

~l 

X 


where x = kr sin 0, 6 being the angle between 
Vi and the direction of a:. A: = 2ix./L Usually only 









DIRECTIVITIES 


141 


the variation of pressure with angle 0 is of in- Other useful facts about these functions are 
terest in which case® given in Table 1. 


Pa = 


Mx) 

X 




Table 1 


The case of the square or rectangular 
bounded radiator is easier to calculate, the re¬ 
sult being 

D ab sin x 

= 5 

Vi X 

where x = {ka 2 ) sin 0 in the plane perpendicular 
to the side a, or x = {kb 2) sin 0 in the plane 
perpendicular to the side b. Again if only the 
variation with angle 0 is needed the first term 
ab/Vi may be omitted as sin x/x = 1 for x = 0. 
A graph of the functions sin x/x and 2J\{x)/x 
= Ai(jc) is shown in Figures 6 and 7. 



X = —^ sin 6. (Circular-plane radiator.) 
k 


The zeros of the sin x/x function corresponding 
to the nulls between lobes come at x = mr, 
while the maxima occur at the roots of the 
equation x = tan x. These were calculated 
first by Schwerd to be x = 4.49, 7.7, 10.9, 
14.1, 17.1, 20.3.' 


e For a table of these Ai(a;) functions see Janke and 
Enide Tables P; also for more details about this integra¬ 
tion see reference 11. 

f The zeros of the Ai(x) functions occur at {x = 3.8, 
7, 10.2, 13.4, 16.4, 19.6, 22 • • • and the maxima at (x = 
5.1, 8.4, 11.6, 14.8, 18, 21.2 • • *. 


Db down 


Formula 


3 

6 

17.8 

23.8 


3 

6 

13.47 

18.24 


Circular 
a = radius 
X = wavelength 

= sin“i 0.258 “ 

a 

-- sin“' 0.350 ~ 

a. 

- sin“i 0.595 "" 1st 

a. 


zero 


= sin“i 0.818 ~ max first lobe 

a. 

■- sin“i 1.11 ~ 2nd zero 
a 

-- sin“i 1.34 “ max 2nd lobe 

a 

■- sin-1 1.62 — 3rd ; 

a 


l zero 


Square 

side of square 


X 

= sin 1 0.446 ~ 

a 

= sin-1 0.605 ~ 

a 


= sin-1 1 00 ~ 1st zero 
a 

= sin-1 43 A jjjgx first lobe 
a 

= sin-1 2.00 ~ 2nd zero 
a 


The directivity function for a rectangle of 
sides a and b in the plane containing the diag¬ 
onal and perpendicular to the face is 

n sin u sin z 
~ u z ’ 

where u = {ka/2) sin O and z = {kb/2) sin 0. In 
the case of the square this reduced to 

It is seen that the side lobes in this plane de¬ 
crease more rapidly, a fact made use of by 
mounting square transducers at an angle of 45° 
to the surface of the water. 














































142 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


For fields close to the transducer, Schwarz- 
schild has attacked the problem in the case of a 
slit wide compared to the wavelength, and many 
authors^-'^® have written on the case of the cir¬ 
cular piston. Most of the solutions involved the 
development of the velocity potential in infinite 
series that do not converge very rapidly so that 
large numbers of terms must be considered in 
the calculations. They have however been able 



Figure 7. ^ as a function of x; x — — sin 0. 

X I 

(Square-plane radiator.) 


to calculate the field down to distances of the 
order of the transducer dimensions, and find 
nulls on the axis close to the transducer in gen¬ 
eral agreement with the results of the infinite 
slit. 

Lobe Suppression in Plane Radiators 

The above equations were developed under 
the mechanism of a plane wave passing through 
a hole, which necessitates the phase and ve¬ 
locity of the wave to be constant over the inte¬ 
gration surface. If the velocity is not constant 
a different directivity pattern results, and is in 
some cases an improvement. When the velocity 
is less around the edges of the transducer than 


in the center, the side lobes are always less in 
magnitude than they are with constant-velocity 
distribution, but the central lobe is generally 
broader. Several methods of calculating the 
sound field from radiators of variable surface 
velocities and phases have been advanced. 

The methods can be generally divided into two 
classes: those that synthesize a velocity distri¬ 
bution from combinations of simple constant 
radiators whose directivities are known, and 
those that use transformations of the type 



that are known or that can be calculated. For 
practical purposes the first method is the most 
useful because transducers are not designed 
with velocities continuously variable but with 
step variation over their surfaces. However, the 
second method has given the patterns for a wide 
variety of distributions which are a valuable 
guide for design and which also give a perspec¬ 
tive to the lobe-suppression art. 

Using the second method, Jones^^ has given 
suppression schemes for nine distributions for 
the circular transducer, and nine for the square 
or linear case, which are plotted in Figures 8 
and 9. The “efficiency” noted there is the ratio 
of the average amplitude of vibration to the 
maximum amplitude. The first or “brute-force” 
method combines constant-velocity surfaces of 
different size to synthesize a step formed ve¬ 
locity distribution, and calculate the resulting 
patterns through combination of the corre¬ 
sponding patterns for each elementary surface. 
For example two circular transducers of ve¬ 
locities ratio 1 to 2 and size ratio 1 to 0.6 would 
be combined as is shown in Figure 10, or analyt¬ 
ically 

R = \i{krsmd) + 2 X (0.6)^ Ai(O.6^rsin0). 

An analogous procedure can be used, of 
course, in the square case. At first thought it 
would seem that the maximum-lobe suppression 
would obtain if the second lobe of one surface 
were placed on the first lobe of the other, but 
this is not the case, because other lobes then 
become important. Several cases using two ve¬ 
locity distributions have been calculated in this 
manner for both circular and square surfaces. 

































DIRECTIVITIES 


143 
























VE 

:loc 

ITY FUh 

ICTK 

)N 






v(U)-i o<usi 







EFFICIENCY 100% 

-1-1-1- ■ 



















10 

8 


































V 

ELOC 

:iTY 

FUl 

SICTH 

^N 






b 

A 

V 

ELO 

CITY 

FU 

NCTl 

ON 







V(U) 

«1-U* 0<U<1 







V(U)-1-U* 0<U<1 






efficiency 

80 % 






0 

EFFICIENCY 

—■ » ..L. 1 

75% 







o .1 .2 .3 .4 .5^.6 .7 .8 .9 I. 


-10 


-20 


-30 


-40 


'N 

. 











V 











T 



OIR 

ECTIVITY 

FUNCTION 
















\ 
























\j 

C\ 










li 

j 

j: 





0 .1 .2 .3 .4 ,5 .6 .7 .8 .9 I. 

U. 


8 12 
a 


16 20 





a 



li. 



a 



U 



a 
























VE 

Locr 

FY 

FUNC 

5TI0 







V(U)«(1-U*)* 0<U< 

EFFICIENCY 35% 

i 













% ,1 .2 ,3 .4 .5 .6 .7 .8 .9 I. 



a 






a 



M. 



Figure 8. Lobe suppression for nine surface velocity functions in the square-plane radiator (Jonesis). 








































































































































































































































































































































































































144 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 
























VE 

o 

o 

ITY 

FUN 

CTIC 

N 






V(A)»i V*<1 







EFFICIENCY iOO X 

1 1 1 1 t 







0 .1 .z ,3 .4 .5 .6 .7 .8 ,9 \. 

U 



0 
























V 

ELOC 

:iTY 

Fur 

^CTK 

DN 




\ 



V(A)«1-V* V*<1 

IFFICIENCY 89^ 





\ 


E 








Ql-1-1-'-^-1-1-1-^-1-L 

0 J .2 .3 .4 .5 .6 .7 .8 .9 I. 

U 














0 .1 ,2 .3 .4 .5 .6 *7 .8 .9 I. 




3 P 0 

Figure 9. Lobe suppression for nine surface velocity functions in the circular-plane radiator (Jones^®). 

















































































































































































































































































































































































































DIRECTIVITIES 


145 


and appear in Figure llA and IIB. Experi¬ 
mentally the velocity ratio of 3 to 1 and diam¬ 
eter ratios 0.6 to 1 have given the most sup¬ 
pressed side lobes so far in the circular type as 
exemplified in Figure 12 patterns of the GA- 
14Z, which includes theoretical pattern as well. 

Lobe suppression can be carried on almost 
indefinitely using additional steps of velocity 
until some sort of gaussian exponential velocity 
distribution is reached. However, the practical 
difficulties (see below) of constructing a crystal 
motor so that the amplitudes and phases of each 


lying on lines perpendicular to this intersection 
line will radiate in the same phase as far as the 
particular directivity plane is concerned.^ 

In other words, in any given directivity plane, 
any plane radiator may be thought of as a line 
radiator that varies in strength from point to 
point as the numbers and strengths of the radi¬ 
ators lying in the radiating plane at right 
angles to the line. With this in mind, lobe sup¬ 
pression may be done by spacing the crystals 
as well as varying their impressed voltages. 
Such a scheme is illustrated in Figure llA 



Figure 10. Method for calculating lobe suppression by velocity combination with the circular-plane 
radiator. 


crystal actually perform as they are specified 
make it useless to shade the velocities in any 
but rather large steps. The relatively small size 
of the transducer thereby usually limits the 
number of these steps to two or three at the 
most. A three-dimensional pattern of a lobe- 
suppressed square-plane radiator is shown in 
Figure 13. 

Lobe Suppression by Crystal Spacing 

Directivities are usually calculated in some 
plane which is normal to the face of the trans¬ 
ducer. The intersection of this plane and the 
transducer face defines a line in which all the 
radiators in the face may be projected without 
changing the calculations, as all the radiators 


and IIB together with the experimental direc¬ 
tivity patterns arising therefrom. This method 
has the disadvantage however of second-order 
main lobes as discussed in Section 4.3.5 when 
the crystal spacing is not small compared to a 
wavelength. See Figure 19A and 19B. 

Phasing 

So far the effect of variations in surface 
velocities only have been discussed. Phase varia¬ 
tions also are important, and both phase and 
velocity variations may be used simultaneously. 
If the radiating surface is uniform in velocity, 
the phase in adjacent lobes differs by 180°. By a 
reciprocal theorem, if the radiator was divided 
into zones that vibrate in amplitudes decreas- 




























































146 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 




Figure 11 A. Effect of step velocity distributions and crystal spacing upon the directivity of a plane 
radiator. 







































































































DIRECTIVITIES 


147 




Figure IIB. Effect of step velocity distributions and crystal spacing upon the directivity of a plane 
radiator. 


































































































148 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


ing in magnitude as the lobes in the pattern of 
the uniform radiator and alternate 180° in 
phase, the pattern should be uniform over a 
certain arc and have no side lobes. This situa¬ 
tion is illustrated in Figure 14. Other effects of 
phase variation can be mentioned. If only two 
symmetrical zones of phasing are used, the 
patterns are similar to Figure 15. The most 
noticeable effect of this type of phasing is in 



Figure 12. Theoretical and experimental di¬ 
rectivity patterns of GA-14Z crystal transducer. 
Solid line: experimental. Broken line: theo¬ 
retical. 


raising the side lobes and eliminating the nulls 
between lobes. If a linear phase shift across the 
radiating surface is used, the main lobe is 
shifted in direction; see Figure 16. Such phas¬ 
ing can be used to train the main lobe elec¬ 
trically while the transducer is fixed. 


^ Directivity Factors 

The directivity factor is the ratio of the total 
energy radiated by a transducer to the energy 


that would be radiated if it radiated its maxi¬ 
mum intensity in all directions. This factor is 
useful in computing the total acoustic power 
from an absolute-intensity calibration made 
upon the principal lobe. It can be computed by 
the formula 

f 

Directivity factor = —-, 

I P- max da 

integrated over a sphere surrounding the 
transducer. Practically, this is a numerical in¬ 
tegration problem and patterns taken in many 
planes must usually be used. Theoretically com¬ 
puted factors for the case of the line, plane- 
circular, and plane-square radiators are given 
in Figure 17. Experimental directivity factors 
require the use of many curves, as most pat¬ 
terns have little symmetry. An example of a 
radiator having reasonable symmetry is shown 
in Figure 18. One showing poor symmetry is 
shown in Figure 13 (see also Section 4.4.2). 


^ ^ ^ Experimental Data 

From the theoretical consideration given 
above it is possible in principle to fashion 
the directivity of a transducer into any desired 
form. Success in this, however, requires the 
radiating surfaces to perform according to the 
prescribed conditions, which leads to one of the 
most difficult problems in the construction of 
transducers. In Sections 3.4, 3.5, and 3.7 the 
nature of this problem was discussed and now 
a few experimental results of nonuniform radi¬ 
ating surfaces will be discussed. Wide varia¬ 
tions in the agreement between theory and ex¬ 
periment are encountered in transducers of dif¬ 
ferent design, and often are encountered in a 
particular unit at different frequencies. These 
departures from theory vary in magnitude from 
negligible to those large enough to render the 
unit useless for its intended purpose. The analy¬ 
sis of these eccentricities can be divided into 
two parts, one treating the main or central lobe, 
and the other treating the side lobes. The most 
important feature of the main lobe aside from 
its absolute intensity is its width, which can be 
defined as the angle subtended by two points on 
























DIRECTIVITIES 


149 














150 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


each side of the center that are 6 db down in 
intensity from the maximum. This theoretical 
beam width for the square- and circular-plane 
radiators with this definition are 

6 = 2 sin-1 0.605— (square), 

a 

and 

6=2 sin-1 0.305- (circular). 

a 

See Table 1, Section 4.3. In general the experi¬ 
mental beam widths are in good agreement with 


usually with good approximation, be predicted 
from the overall dimensions of the transducer. 

The side lobes, however, being diffraction 
patterns, are much more sensitive to surface 
variations in phase and velocity. Agreement 
with theory in them is rarer than in the central 
lobe, a situation that is often troublesome in 
applications. With uniform velocity and phase, 
theory predicts, for example with the square- 
plane radiator, that the secondary lobes occur 
at regular angle intervals with intensities regu¬ 
larly decreasing 13 db, 18 db, 21.5 db, etc. The 
angle intervals vary regularly with frequency. 



Figure 14, The reciprocal relationship between the 
directivity function in a square-plane radiator. 



surface velocity function and the corresponding 


theory even when the accompanying side lobes 
are in very poor agreement. Calculations of pat¬ 
terns using more or less heterogeneous surface- 
velocity distributions show that the main-lobe 
width is less dependent upon the velocity dis¬ 
tributions than are the side lobes. Only with 
ordered surface-velocity variations is the main 
lobe affected, and even then not as much as the 
side lobe. The width of the central lobe can thus. 


but the intensity ratios remain the same. Fig¬ 
ure 21 shows a set of patterns for such a 
radiator in which no pattern shows any regular 
change in secondary-lobe angles, nor do these 
lobes remain the same in relative intensity. Not 
much is known about the causes of such erratic 
patterns, because while a given surface distri¬ 
bution may cause definite patterns, the converse 
that a given pattern mav be caused by one and 



























































DIRECTIVITIES 


151 


only one surface condition is not generally true. 
A discussion of a few factors known about the 
effect of surface condition upon directivities can 
throw a little light upon the situation. 


Measurable Cause and Effect Factors 

We have seen that a radiator whose center is 
stronger than its rim will radiate less energy 
in the side lobes. The pattern of such a radiator 



ator in which the phase of the velocity at the 
ends is shifted 90°. The absolute magnitude of 
the velocity is constant throughout. 

(the GA-14Z) is shown in Figure 12, both 
theoretically and experimentally. The side lobes 
of the experimental pattern are generally much 
less than those of a uniform radiator, but they 
are not congruent to the theoretical pattern, 
nor are they symmetrical. This effect could 
obtain only from a nonsymmetrical radiator. 
Conversely, a radiator whose rim is stronger 
than its center will radiate more energy in the 
side lobes. See Figures llA and IIB. Also, a 


radiator whose rim is out of phase with its 
center will radiate more energy in the side 
lobes. See Figure 15, 



Figure 16. Shifting of the main lobe by a 
linear phase variation over the length of a line 
of point radiators. Phase shifted 30° per point 
radiator. 



Figure 17. Theoretical directivity factors of 
three types of sound radiators. 


Another striking property of this kind of 
radiator is the disappearance of nulls between 









































































































152 


PROPERTIES OE ASSEMBLED CRYSTAL TRANSDUCERS 


side lobes. Notice that most of the experimental 
patterns illustrated in this section have many 
nulls eliminated between side lobes, a fact that 
indicates nonuniform phasing in most trans¬ 
ducers. 

Frequently the crystal matrix of a motor is 
divided into units of two, three, or four crystals 
forming a block. These blocks are spaced from 
each other of the order of to in. With the 
blocks operating under oil, this space may pre¬ 
sent a radiating element 180° out of phase with 
the crystal surface because the sides of the 
crystal blocks are vibrating 180° out of phase 
with the ends, and this motion is transmitted 
to the oil cavity between crystal elements. In 
addition to this effect, there is another one due 
to the spacing. If n point radiators are put in a 
line distant from each other by d, the directivity 
is given by 

sin 

R = - 

n sin 

instead of 



As n gets large and d gets correspondingly 
small, these two formulas approach each other. 
The first formula will give a second maximum 
when the nd/X sin 6 reaches n or when sin 9 = 
X/d.lf d = X then there should be a second max¬ 
imum equaling the central lobe at 0 = 90°. 
Finite spacing between crystals when working 
under oil should then give two effects, one a 
smear effect upon the side lobes, the other a 
second large maximum at high frequencies. An 
example of each effect is shown in Figures 19 
and 20. Figure 19A shows patterns at several 
frequencies of a transducer in which the crystal 
spacings are appreciable at different frequen¬ 
cies. At the higher frequencies, the second large 
maximum is evident. This effect is similar to 
the second-order spectrum noticed with optical 
gratings. Another example of this effect can 
be seen in the GD22 transducer. This trans¬ 
ducer is constructed with 24 ADP crystals 


Cycle-Welded directly to the rubber window 
with the technique described in Section 8.5.6. 
For this discussion it can be considered as a 
grating-type radiator in which the distances 
between individual radiators is as large as the 
radiators themselves. In the center of each 
diagram of Figure 19B is a chart showing the 
motor layout. The distances between centers 
along one axis is twice that along the other axis. 
Figure 19B illustrates a pattern taken in these 
two directions. Figure 19B-a is the pattern in 
the direction of smaller spacing and no large 
secondary maxima are evident. In Figure 19B-b 
this second-order effect is just beginning to 
appear. At higher frequencies the large lobes 
on the side increase in size and decrease their 
angle with the zero axis. The widths of the 
main lobes of both are the same because the 
motor is square. Figure 19B-c illustrates the ef¬ 
fect of the intercrystal spacings in smearing 
the side lobes. Patterns were taken with the 
motor air-filled and oil-filled. Figure 19B-C 
shows immediately the effect of filling the cavity 
with oil which could be due to intercrystal vi¬ 
bration or to overall cavity modes. That the 
effect is due to the intercrystal spacings is 
shown by Figure 19B-d, which is a pattern 
taken after each crystal had been surrounded 
with foam rubber (the motor still working in 
oil). This pattern is nearly as clear as Figure 
19B-b. 

The transducer case has its effect also upon 
the directivity. If the window in the case is a 
poor transmitter, standing waves are set up 
inside the case which obscure somewhat the 
connection between the motor and the water. 
Figure 21 shows a contrast of the patterns of 
the same motor in two cases, one of which was 
a simple qc-rubber case. In this comparison, the 
more transparent case greatly improves the 
patterns. 


“ APPLICATIONS OF RECIPROCITY 

In this section the reciprocity principle is 
applied to a generalized transducer, and certain 
conclusions are drawn concerning the sound 
field. For this purpose the transducer is as¬ 
sumed to have certain ideal properties such as 






APPLICATIONS OF RECIPROCITY 


153 


uniform phase and amplitude over plane arrays, 
and the conditions requisite for reciprocity 
(linear, passive, bilateral). Fortunately the 
conclusions reached are functions only of the 


vided the radiating surface fulfills require¬ 
ments. 

As a matter of fact, 45° Y-cut RS and 45° 
Z-cut ADP obey reciprocity very well. There is 



Figure 18. Three-dimensional patterns of symmetrically di’iven double plane radiator. 


radiating surface, its geometry, amplitude, 
phase, etc., but not of the nature of the trans¬ 
ducing material (crystal, nickel tubing, etc.). 
Thus these conclusions apply equally well to 
transducers which fail to obey reciprocity, pro- 


some question of the linearity of glued joints 
under large stress, and finished transducers 
may fail to obey reciprocity under very high- 
power operation. If this does occur it is rela¬ 
tively rare. 













154 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


LoIdc Suppression 

When an array is lobe-suppressed it is fre¬ 
quently necessary to drive one portion at 
greater amplitude than another. If the same 


Section 4.8), and it is apparent that the total 
power radiated by the lobe-suppressed array 
must be less than that radiated by the uniform 
array. The quantity of interest in a transmitter 
is not the total power radiated, but the acoustic 



Figure 19A. Second-order maxima in the directivity pattern of a transducer in which the crystal 
spacing’s are not small compared to the wavelength radiated. 


array were not lobe-suppressed all elements 
could be driven equally. When operating as a 
transmitter a limitation on output power will 
be imposed by the hardest driven area (see 


pressure produced in some direction (usually 
the direction of maximum response). By lobe¬ 
suppressing, power which was previously radi¬ 
ated into undesired side lobes is directed into 



















































APPLICATIONS OF RECIPROCITY 


155 


the main lobe; this raises the pressure in that 
direction and helps to make up for the dimin¬ 
ished power output. On the other hand, lobe sup¬ 
pression always causes the main lobe to become 
a little wider, and this tends to diminish the 



pressure. It is not possible to weigh the impor¬ 
tance of these qualitatively in order to learn 
the net effect. In this section the result is ob¬ 
tained by use of reciprocity. 

Consider a transducer composed of a large 


number of identical active elements closely and 
uniformly spaced in a large plane array and 
all connected in phase. Let any element be rep¬ 
resented by a generalized transmission T (see 
Figure A, page 158). 



If each element has radiating area a the 
radiation impedance may be represented by 
some quantity aZ^; (see Figure B, page 158). Z,. 
will depend upon the geometry, the nature of 
the medium, etc., and is not strictly the same 



Figure 19B. Directivity patterns of the GD22Z transducer, at 60 kc, showing the effects of: crystal 
spacings, oil versus air inside the transducer case, and foam rubber inserts between the crystals when the 
case is oil filled. 















































































156 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



Figure 20. Directivity patterns of the FG8Z-3 transducer showing the effect of foam rubber inserts 
between the crystals in an oil filled transducer at different frequencies. Solid line: with inserts. Broken 
line: without inserts. 


for all the elements. However, it is usual in 
calculating the radiation field to make certain 
assumptions tantamount to making a con¬ 
stant for all elements. 

A normally incident sound wave introduces 


a voltage in this circuit proportional to the 
area a. (Refer to Figure C, page 158.) 

If, now, the electric terminals are shorted 
some current Iq will flow. If the normally inci¬ 
dent wave had a free-field pressure of 1 dyne 































































































APPLICATIONS OF RECIPROCITY 


157 



Figure 21. Directivity patterns of the CQ6Z6-4 crystal motor showing a contrast between the effects 
of neoprene and oc-rubber cases filled with oil. Solid line: gc-case. Broken line: neoprene case. 













































































































































158 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


per sq cm, we term Iq the short-circuit receiver 
response of the element. 

Lobe suppression is achieved by connecting 
some of these unit elements in series, groups 
of the series elements then being in parallel 



with each other and with other individual ele¬ 
ments, all in phase. It is easily shown that if 
any number of these elements are put in series 
the short-circuit current which flows from the 
group is the same as the current from a single 
element. 



Assume that there is, in some plane array, 
a number N of these elements. If this array 
were connected all in parallel for no lobe sup¬ 
pression then all N elements would contribute 
current h, and the short-circuit receiver re¬ 
sponse of the whole transducer would be N/o. 



Now suppose that the elements in the array 
were reconnected for lobe suppression. This 
may involve all kinds of series-parallel arrange¬ 
ments, all in phase, depending upon the lobe- 
suppression scheme. There may be: 
n-i elements simply in parallel, 

Uo elements paired in series, the pairs in par¬ 
allel. 


elements in series-triplets, the triplets in 
parallel, etc., subject to the limitation: 

Ui + Jiz = N. 

However each pair will contribute only /o, 
each triplet h, etc. Consequently a great many 
crystals might be considered wasted. The total 
short-circuit receiver response of the newly con¬ 
nected array is only 

(Wl + 1^2 + + • • • ) 

and is always less than NIq. Thus the effect of 
lobe suppression is to reduce the short-circuit 
receiver response of the transducer by a factor 

N 

However the reciprocity principle states 
[Section 4.2.3, equation (21)]: 


Where, if R represents the short-circuit re¬ 
ceiver response, T represents the pressure pro¬ 
duced in the water at some distance for 1 v 
applied to the transducer (the so-called con¬ 
stant-voltage transmitter response). 

If subscript 1 indicates the original uni¬ 
formly driven array, and subscript 2 indicates 
the lobe-suppressed array, then 



and it was shown above that this is equal to: 

T, = + ■ ■ • (65) 

and we see that To will always be less than Ti. 

This is the expression sought, giving the dim¬ 
inution in intensity radiated caused by lobe 
suppression. As given here it is for the same 
voltage applied to the two arrays. It would be 
a simple matter to compute the impedance 
change resulting from the lobe suppression, and 
then express the change in intensity for 1-w 
input to each array. However the maximum 
power output at resonance is fixed by cavitation 
of the single elements, and away from reso¬ 
nance by voltage breakdown of the single ele¬ 
ments; this will mean a maximum allowable 
voltage at any frequency, the same for both 




















APPLICATIONS OF RECIPROCITY 


159 


arrays, and the expression above gives the re¬ 
duction in maximum possible intensity result¬ 
ing from the lobe suppression. 

Stated in terms of numbers of elements, this 
expression is not as might be desired; it can be 
restated in terms of areas as follows: 

The N elements, each of area a, when closely 
packed, occupy a total area A. The Ui elements 
occupy area Ai, the no elements occupy area Ao, 
etc. Thus we may picture the array as a group 
of zones: one zone of area Ai is driven at unit 
amplitude, another of area Ao is driven at V 2 
amplitude, etc. Then 

Ai IA2 + + • • • 

A 

This form of the expression emphasizes an 
interesting feature: within our assumption that 
the radiation impedance is uniform over the 
array, the reduction is independent of the 
shapes of the zones and depends only on their 
areas. 

It is interesting to evaluate To/Ti for two 
particularly common lobe-suppression schemes 
applied to circular arrays: 

1. An inner disk, whose diameter is 0.5 of 
the array diameter, is driven at unit amplitude; 
the remaining annulus is driven at V 2 ampli¬ 
tude. Theoretically the first side lobe is down 
22 db. 

2. An inner disk, whose diameter is 0.61 of 
the array diameter, is driven at unit amplitude; 
the remaining annulus is driven at Ys ampli¬ 
tude. The first side lobe is down 28 db. 

The expression for To/Ti gives the reduction 
for scheme 1 as 4.1 db, a quite sizable reduction. 
For scheme 2 the reduction is 4.7 db. If one is 
reconciled to such losses for the sake of lobe 
suppression, then 4.7 is negligibly worse than 
4.1 db, and 2 is the preferable scheme. 




“ Directivity Index 

Measurements of directivity index are re¬ 
quired in order to obtain the efficiency of a 
transducer, and it is unfortunate that this is the 
most difficult and least accurate of all calibra¬ 
tion tasks. In the present state of the art very 
great care is required to obtain an accuracy of 


±1 db, and errors of ±2 or 3 db are much more 
usual. For this reason and for obvious theoret¬ 
ical reasons it would be desirable to obtain an 
expression for the directivity factor and index. 

If one is willing to make the usual assump¬ 
tions of uniform loading, uniform (or at least 
analytic) phase and amplitude distributions, 
infinite baffle, etc., one can write an integral 
expression for the directivity factor. For sim¬ 
ple radiators this expression is tedious but may 
be evaluated. A simple arithmetic expression is 
obtained below, fairly rigorous for any array, 
and evaluable for any array for which the 
above assumptions are justified. As in Section 
4.4.1 the proof uses the reciprocity principle, 
but the results depend only on the nature of 
the radiating surface, and are applicable to 
transducers which disobey reciprocity. 

Consider a radiating surface of arbitrary size 
and shape driven with reasonably simple phase 

o - nn 

ELECTRICAL 

O--O 

Figure D. 

and amplitude distributions. (Refer to Figure 
D.) Such a transducer may be represented by a 




generalized transmission T. (All Z’s may be 
complex.) 

If the area of the radiating surface is A and 



the transducer is immersed in a medium of 
characteristic impedance qc the radiation im¬ 
pedance may be represented as Zj, = gcAig + 




















160 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


jh). The imaginary term is trivial, and since it 
might be included in Zo it is omitted, and the 
equivalent circuit becomes that shown in Fig¬ 
ure E. 

Let a plane wave whose free-held pressure is 
1 dyne per sq cm be incident from the direction 
of maximum response. This introduces a voltage 
proportional to the area shown in Figure F. 

Both g and y are unknown except for certain 
very simple radiators. Furthermore g and y 
will vary from point-to-point over the surface, 
and the values inserted here are some average 
over the surface. For simple plane sources con¬ 
taining a small number of zones of constant 
phase and amplitude, the averaging process 
would be easy if good approximations were 
acceptable. For a completely arbitrary phase 
and amplitude distribution, one might wishfully 
hope for some average, at least in principle, if 
the distributions were not too badly nonanalytic. 

One may then calculate the current I which 
would how if the electric terminals were short- 
circuited. The current will be proportional to 
yA, and may be written 



where is a transfer impedance, and is a func¬ 
tion of Zi, Zo, and Z^. 

Since we are concerned hnally with the radi¬ 
ating surface we may assume, without loss of 
generality, that the T is passive, linear, and 
bilateral. Then, by invoking the reciprocity of 
such electric networks, we may write the cur¬ 
rent 7^ which would how through if the 
transducer were driven as a transmitter by 1 v 
applied to the electric terminals: 



From this we may calculate the electric power 
expended in the radiation impedance: 

Pr — \IrV 

Holding this in abeyance, we now invoke the 
acoustic reciprocity principle; for this we must 
assume that the medium is also passive, linear, 
and bilateral; conditions which water satishes. 
We obtain the pressure p (dynes per sq cm) in 
the medium at distance r cm, in the direction 


of maximum response for 1 v applied as a trans¬ 
mitter. The reciprocity principle relates this to 
the short-circuit receiver current 7 [Section 
4.2.3, equation (41) ]. 



where / is the frequency in cycles. 

Now if pressure p were radiated equally in 
all directions the acoustic power output for 1 v 
applied would be 

P, =P^ . 47rr^ 

pc 

Actually p is not radiated in all directions, 
and we know that the total power input is Pj^. 
Consequently the directivity factor D is given 
by 



_ pcAg . pc4r- i Zr I" 

j Zr 12 p-fA-K-pA -’ 



This is the expression for D; the directivity 
index is 10 log D. 

For the vast majority of transducers, g and 
Y“ and their ratio are unknown. For an infinite 
plane, uniformly driven in phase and amplitude, 
g = 1 and y = 2. For a point g = D and y = 1; 
no other surfaces (except perhaps a sphere) 
are known. It is not surprising that this trick 
with reciprocity has not introduced anything 
new to radiation theory. However the function 
still has great use. 

Experience indicates that uniformly driven 
plane sources have directivity patterns and re¬ 
sponses in remarkably good agreement with 
those predicted by the theory requiring infinite 
baffle, etc.; this seems to be true even for sources 
whose dimensions are as small as 1 wavelength. 
Consequently it is a fair approximation to set 
g/y- = 1/4 for such sources. This approximation 
is exactly equivalent to the simplifying assump¬ 
tions usually made in computing patterns, etc. 
Thus the numerical value of D obtained is ex¬ 
actly the same as would be obtained from the 





COMPLEX IMPEDANCE 


161 


integral evaluation when the integral can be 
evaluated at all. For all presently practical pur¬ 
poses, then, we may write: 


In numerous checks this function has been 
found to agree with integral evaluations and 
with experimental data well within the experi¬ 
mental error. 

Criteria may be established by the accuracy 
of this function. 

1. We know that for large plane A, g/y- = 14 . 
We know also that for very small A, D =zl, and 
consequently g/y- -» 0. We know also that the 
function works fairly well for A>P. It is likely 
that the approximation goes bad rapidly for 
A<1\ 

2. If the observed directivity patterns agree 
with those predicted by the simple theory using 
infinite baffles, etc., then the approximation 
g/y 2 — is justified. If the patterns disagree 
there is no way of evaluating D theoretically; 
it can be obtained only by numerical integra¬ 
tion of the observed patterns. 

In conclusion, note that g/y- = for A in¬ 
finite and plane, and is equal to 0 for A = 0. 
We may speculate on the behavior elsewhere, 
and, in particular, wonder if the connection is 
monotonic with A. 


4 - COMPLEX IMPEDANCE 

When electrical measurements are made at 
the terminals of a single transducer without 
simultaneous sound-field measurements the only 
quantity directly observable (except thermal 
noise) is the complex impedance which may be 
obtained as a function of frequency, power, etc. 
Additional information, such as efficiency and 
response, can be obtained only if the equivalent 
circuit is known; the accuracy with which such 
information may be deduced depends upon the 
completeness and accuracy of the assumed cir¬ 
cuit and is always subject to doubt. At best an 
equivalent circuit is an approximation, and 
there always remains a considerable burden of 
proof that the approximation is adequate. In 
fact University of California Division of War 


Research [UCDWR] feels that there has yet to 
be built the unit whose electroacoustic efficiency 
can be obtained by electrical measurements. 


Motional Impedance 

One may measure the complex impedance of 
a transducer in water (or a motor in air) as a 
function of frequency. In principle one might 
measure the complex impedance of the same 
unit as a function of frequency when mechan¬ 
ical motion was blocked. The vector difference 
between these is called the motional imped- 
ance.^^ If the real and imaginary parts of the 
motional impedance are plotted linearly as Car¬ 
tesian coordinates with the same scales, the 
resulting locus as a function of frequency is, 
ideally, a circle passing through the origin. The 
diameter passing through the origin cuts the 
circle at the point belonging to the resonant 
frequency; the diameter at right angles to this 
one cuts the circle at two points belonging to 
the frequencies at which the mechanical re¬ 
sponse is down 3 db (i.e., response for constant- 
voltage drive in Mason circuit). 

While it is possible to determine the blocked 
impedance of such devices as telephone dia¬ 
phragms, this is certainly not possible with 
crystal transducers. The blocked impedance is 
estimated by interpolating a smooth curve con¬ 
necting the resistance well above and well below 
resonance, and the same for the reactance. The 
difference between the observed resistance and 
reactance and the interpolated resistance and 
reactance are assumed to be the components of 
the motional impedance. Then the efficiency at 
any frequency is estimated to be the ratio of the 
motional resistance to the total resistance. 

This procedure obviously embodies a great 
many assumptions. The method of interpolation 
tacitly assumes that the mechanical branch is 
singly resonant in the interpolation region; 
quite frequently this is not true, and the mo¬ 
tional-impedance locus is several circles super¬ 
imposed. For reasons of this kind it is not un¬ 
common to find that several different efficiencies 
could be obtained. 

At best the efficiency so obtained gives the 
efficiency of conversion of electrical energy to 




162 


PROPERTIES OE ASSEMBLED CRYSTAL TRANSDUCERS 


mechanical. One can imagine situations in 
which it does not even do that. For example: 
suppose the equivalent circuit of a transducer 
were as shown in Figure G. The resistance 
shunting the mechanical transformer is me¬ 
chanical, yet it would be lumped with the elec¬ 
tric impedance and would not be included in the 
motional resistance. 

The electro-mechanical efficiency of Y-cut RS 
and Z-cut ADP is very high, so that if reason¬ 
able care is taken to reduce electrical losses 
(e.g., series resistance of wires, dielectric loss) 
the overall efficiency of conversion to mechan¬ 
ical energy should be very high. It is a poor 



transducer which fails in this respect. On the 
other hand, the largest component of the mo¬ 
tional impedance may be internal losses such as 
occur in glued joints; a high-motional imped¬ 
ance efficiency might be associated with a very 
low-acoustic efficiency. For this reason mo¬ 
tional-impedance efficiency is virtually useless, 
and is now going out of use with crystal trans¬ 
ducers. 

Absolute Admittance 

By the absolute admittance we mean the 
length of the admittance vector: the square root 
of the sum of the squares of the conductance 
and susceptance. 

This quantity is easily obtained with one or 
two meters and a known resistor, usually by 
measuring the current into a transducer when 
driven by a constant-voltage source. The ease 
of measurement is the chief reason for using 
the quantity, since it tells one much less than 
does a complex measurement. 

Remote from resonance the admittance of 
crystals in air behaves like that of a fixed con¬ 
denser. This contains the admittance of Co in 
the Mason circuit, and also of the mechanical 


arm which, remote from resonance, behaves 
like a condenser. Thus the observed admittance 
below resonance is greater than that of Co alone 
(by about 9 per cent for Y-cut RS and Z-cut 
ADP) and this correction must be made when 
computing the dielectric constant (see Section 
3.2.3). 

At resonance the admittance of the mechan¬ 
ical arm rises to that of the load impedance; 
for free crystals in air a Q of 1,000 or more is 
common, and the admittance at resonance rises 
many decibels.^ For high Q, care must be taken 
to be sure that the series resistor used to deter¬ 
mine current is not the limiting resistance in 
the circuit. 

Slightly above resonance, electrical antireso¬ 
nance occurs; the admittance falls to a very low 
value determined by the Q of the circuit. 

Above electrical antiresonance the mechan¬ 
ical branch is inductive, so the admittance re¬ 
mains less than that of Co alone. However at 
mechanical antiresonance the admittance of the 
mechanical arm is zero, and the net admittance 
is that of Co alone. 

Above mechanical antiresonance the mechan¬ 
ical branch is capacitative and the net admit¬ 
tance is greater than Co, rising to the next 
resonance. 

Measurements of absolute admittance of sin¬ 
gle crystals or groups of crystals are used to 
determine crystal constants, glue losses, fre¬ 
quency shifts caused by added structure, and 
so forth. Such data are readily obtained and are 
one of the best research and design tools. When 
diagnosing a finished transducer which fails to 
meet expectations it is very helpful to have 
available admittance data on the bare dry 
motor; this helps in deciding whether the trou¬ 
ble is inherent in the motor itself or in the case, 
cavities, window, etc. It is well for a transducer 
laboratory to take admittance data on subassem¬ 
blies as a routine practice. 

Little use can be made of the absolute admit¬ 
tance of finished transducers, particularly oil- 
filled units, either in air or in water. In water 
the Q’s are so low that the admittance, even 
that of efficient units, departs little from that 
of a fixed condenser and all units look pretty 

- By decibels here is meant 20 log current for con¬ 
stant voltage. 











COMPLEX IMPEDANCE 


163 


much alike. In air the radiating face sees an 
impedance very different from that in water; 
standing waves are set up, with accompanying 
swings of admittance, which do not exist in 
water. One usually observes many erratic peaks 
and dips of admittance which are not mean¬ 
ingful. 

‘ Two-Terminal Impedance 

A transducer may be regarded as a two- 
terminal system, electrically, with any shield 
tied to ground or floating and lead-to-shield 
effects ignored. Calibration stations usually re¬ 
port complex impedances this way unless more 
elaborate data are requested. Such data are 
necessary to an understanding of a transducer’s 
performance and to design of associated elec¬ 
tronic equipment. The impedances to be ex¬ 
pected of various units are discussed in Section 
4.9; here we merely note some general princi¬ 
ples and cautions to be observed. 

Transducers are usually operated balanced to 
ground to minimize noise troubles, etc. It is 
imperative that two-terminal impedances be 
taken balanced to ground if the unit is to oper¬ 
ate balanced. Otherwise the effects of lead-to- 
shield impedances are improperly portrayed. 
However the majority of impedance bridges are 
inherently unbalanced, as are standards for use 
with bridges. This requires the use of isolation 
transformers, and great care must be taken to 
select acceptable transformers. 

If, for some reason, a transducer is to be 
operated unbalanced the impedance should be 
measured under identical balance conditions, 
even to the choice of which transducer lead is 
closer to ground. 

Tuning coils are often put between the trans¬ 
ducer and its cable. Occasionally the compo¬ 
nents are such that the net-series inductance of 
transducer plus coils can parallel-resonate with 
the shunt capacity of the cable. This causes the 
impedance seen at the top of the cable to go 
through gyrations. If the test cable is not iden¬ 
tical with the cable to be used on the unit, exper¬ 
imental errors make it impossible to calculate 
the cable out of the observed impedance. Since 
this resonance with the cable is never desirable 
it should be avoided. 


No bridge now available is capable of meas¬ 
uring accurately over the range of Q and im¬ 
pedance presented by a wide variety of trans¬ 
ducers. Large units, at resonance, may look 
like perhaps 100 ohms at Q = 5; if tuned with 
coils, Q = 0. Small units may look like 100,000 
ohms at resonance. Away from resonance Q may 
be many hundred. To cover all situations a 
bridge may be called upon to measure Q any¬ 
where from zero to 100, reactances of both 
algebraic sign, and magnitudes of impedance 
from a few ohms to a megohm. Since no one 
bridge can do all this, care must be taken to 
select a bridge suitable for a given transducer. 

The complex impedance of itself does not 
allow a diagnosis of transducer behavior; to¬ 
gether with frequency-response data and direc¬ 
tivity patterns it is an important tool. If a 
transducer misbehaves this is likely to be re¬ 
flected in the impedance. For example, low effi¬ 
ciency usually results in an abnormally low 
series resistance, often with no discernible reso¬ 
nance. Spurious vibrations such as harm pat¬ 
terns often show as extra resonant peaks in 
the series resistance. High Q resonances which 
cause sharp holes in frequency response usually 
show as notches in the series resistance. 

The reactance is dominated by Co and is 
not as greatly affected by mechanical effects. 
Usually the reactance is substantially that of 
Co and may not depart noticeably except at 
places where the series resistance rises to im¬ 
portant peaks. Any erratic behavior of react¬ 
ance not accompanied by understandable re¬ 
sistance changes is most likely an indication of 
trouble in the measurement setup, but may 
represent electrical quirks such as saturation 
of tuning coils. 

4.5. t Four-Terminal Impedances 

Unless one side of a transducer is tied to 
ground the circuit is actually at least three- 
terminal. Furthermore the backing plate may 
not be grounded and may be used as a fourth 
terminal. Because of capacity coupling there is 
always some complex impedance between any 
pair of these terminals. Transducers are al¬ 
ways small compared with an electric wave¬ 
length so that all impedances, except perhaps 




164 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


that between the crystal leads, will usually 
have capacitative reactances; the resistance is 
usually dominated by dielectric losses rather 
than by series resistance. 

It is exceedingly difficult to determine all six 
impedances of a four-terminal mesh by measure¬ 
ments on a simple two-terminal bridge. For this 
the techniques discussed in Section 9.1 are 
indicated. 

Some generalizations may be made on the 
basis of work done by UCDWR: The most im¬ 
portant stray capacities are those from each 
crystal lead to the backing plate. Usually the 
clearances from crystals to external case are 
much greater with consequently lower capacity. 
Even in transducers whose crystals are sepa¬ 
rated from a metal backing plate by only Kc-in. 
porcelain, the stray capacities to the plate 
change the apparent value of Co by perhaps 
10 per cent. If the crystals are attached directly 
to metal these capacities are much greater and 
such practice should be avoided. Much more 
extensive investigation is required, but it ap¬ 
pears that these strays are one of the many 
small factors contributing to possible poor be¬ 
havior ; they are not usually in themselves dom¬ 
inating troubles, but efforts should be made to 
minimize them. 

Such measurements as are available indicate 
that some transducer materials, notably Cor- 
prene, have high dielectric losses; if the stray 
capacities are small these losses are negligible, 
but care should be taken to minimize electric 
fields occurring across Corprene and similar dis¬ 
sipative materials. 

The most serious effect of the stray capacities 
is in raising the electrical Q (Q^) of a trans¬ 
ducer and thus narrowing its operating band. 
It is apparent that if there are stray capacities 
from each crystal lead to ground the effect on 
Qj^ will be lessened by balanced operation as 
compared with one side tied to ground. 

4 STEADY-STATE RESPONSES 

Unless otherwise stated, response usually is 
taken to mean response versus frequency for 
some constant-drive condition. Steady-state re¬ 
sponse means that all conditions (both fre¬ 
quency and drive amplitude) are held constant 


for sufficient time to insure decay of transient 
conditions. Under these conditions it is a simple 
matter to calculate the expected response, at 
any frequency and level, to the Mason approxi¬ 
mation, provided the termination impedances 
and directivity index are known. This is dis¬ 
cussed in Section 4.9. In this section we consider 
the utility of the various response curves and 
their general nature. 

Responses are usually expressed in terms of 
constant voltage, constant pressure, constant 
current, etc. In the course of calibration the 
voltage, pressure, or current is rarely constant 
with frequency and is rarely the unit value re¬ 
ported. Corrections to the observed data are 
applied in order to report as if the quantity 
were unit and constant. In most transducers 
this is trivial, but some transducers, notably 
X-cut RS imits, are nonlinear. In such units the 
response in service may be different from that 
indicated. This is particularly true since cali¬ 
brations are made at low level and most trans¬ 
mitters are operated at high level, perhaps as 
much as 60 or 80 db higher. 

Transmitter responses usually give the pres¬ 
sure at 1 m. The distance may be measured 
from the center of the acoustic window, the 
center of the crystal array, or (more usual) 
the geometric center of the body. In some cases 
it may be important to state this distance 
clearly. The responses are nearly always meas¬ 
ured at greater separation in order to be in the 
region where inverse square law is obeyed and 
to make the units better approximations to 
dimensionless sources. The data are then calcu¬ 
lated back to 1 m by assuming inverse square 
law. Actually 1 m is too close for most units; 
it is within the induction field, and the true 
pressure at that distance may be very different 
from the reported response. 

^ Constant-Voltage Transmitter 

Response 

This is the response versus frequency for 
constant voltage applied to the transducer ter¬ 
minals. The voltage is usually reported as 1 v 
and the response is reported as pressure in deci¬ 
bels above 1 dyne per sq cm at 1-m distance. 

This response is one of the most frequently 



STEADY-STATE RESPONSES 


165 


encountered; it is easily obtained, does not re¬ 
quire any corrections for the cable (unless IR 
drop in the copper is significant), and is par¬ 
ticularly useful to designers. Actually no trans¬ 
ducer is operated out of a zero impedance 
source, so that this is not a true picture of the 
frequency response to be expected when the 
transducer is put in service. If the transducer 
is limited by voltage breakdown this curve gives 
a good picture of the maximum possible output 
versus frequency. 

If no tuning coil is used in the transducer 
this response always peaks at or very near the 
transducer’s mechanical resonance unless the 
transducer is so inefficient that no discernible 
resonance occurs. If a series tuning coil is used 
the response peaks at the frequency at which 
the coil cancels the transducer’s reactance; if 
this frequency is not the transducer’s own reso¬ 
nance a second, lower, peak or plateau may be 
observed. When operating with a coil it should 
be remembered that the voltage across the 
transducer terminals at the peak response is 
Q times the cable voltage, this factor may be 
large, particularly if the reactance is cancelled 
away from the transducer’s resonance, and 
transducers are often damaged by failure to 
observe due caution. 

Since the presence of Co has no effect when 
constant voltage is applied, the response curve 
is governed entirely by the behavior of the me¬ 
chanical branch of the equivalent circuit; near 
resonance this is symmetric, so the response 
curve falls symmetrically on either side of 
resonance. 

The maximum value attained depends, among 
other things, on the thickness of the crystals. 
A fictitious improvement appears to be obtained 
if a greater number of thinner crystals is used, 
but the impedance changes and the increased 
pressure is obtained by proportionately greater 
power consumption. This factor must be borne 
in mind when comparing transducers on the 
basis of constant-voltage transmitter response. 


Coiistaiit-Ciirrent Transmitter 

This is the response versus frequency for 
constant-current input to the transducer ter¬ 


minals. It is usually reported as pressure in db 
above 1 dyne per sq cm at 1 m for 1 amp input. 

This response is often reported but is more 
difficult than constant voltage because of the 
cable. Since cables are changed frequently it is 
wise to consider them separately and to report 
the behavior of the transducer proper (as if 
measured at the terminals to which the cable 
would be attached) with the cable removed. 
However, one must have a cable to lead from 
the underwater transducer to the above-water 
equipment, so the cable must be calculated out 
of the data. Usually the current is measured at 
the top of the cable and the correction for the 
current which flows in the cable capacity is 
obtained from impedance data. Some high- 
impedance transducers may have lower admit¬ 
tance than their cables, and serious errors arise 
in the cable correction. For such units the con¬ 
stant-voltage transmitter response is more 
likely to be correct. 

This response is not as convenient for design¬ 
ers as the constant voltage because the equiva¬ 
lent-circuit arithmetic is more tedious. Further¬ 
more, stray capacities to ground affect the 
constant current but not the constant-voltage 
response. It is easier to learn about the strays 
from impedance data. 

Transducers are often operated out of pen¬ 
tode amplifiers which approximate constant- 
current sources (within limits) so the constant- 
current response may closely resemble the re¬ 
sponse in service for such amplifiers. This 
similarity must be tempered with many reser¬ 
vations concerning overloading, distortion, etc., 
and should not be relied upon indiscriminately. 

For an efficient transducer the acoustic power 
radiated at constant-current drive is propor¬ 
tional to the series resistance, peaking near 
resonance and falling, nearly symmetric, on 
either side. The response is determined not only 
by the mechanical branch of the equivalent cir¬ 
cuit, but also by Co; both the electrical and the 
mechanical Q’s enter the calculations. The di¬ 
rectivity index changes with frequency, usually 
between 3 and 6 db per octave, so the response 
of an efficient unit is just the shape of the series 
resistance with that slope added. This response 
is not affected by the presence or absence of 
series tuning coils. 



166 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


Like the constant-voltage response, a ficti¬ 
tious improvement is obtained by use of thinner 
crystals; the series resistances should be consid¬ 
ered when comparing transducers on the basis 
of constant-current response. 

^ Constant-Power Transmitter Response 

This is the frequency response, usually ex¬ 
pressed as pressure in db above 1 dyne per sq 
cm at 1 m, for constant power (1 w) expended 
in the transducer. To obtain this one must have 
not only constant voltage or constant-current 
response data but also the complex impedance; 
cable corrections must be made as for constant- 
current data, and it may be necessary to cor¬ 
rect for the cable dissipation. This involves a 
fair amount of work, and this response is too 
rarely reported. Its chief value is to the de¬ 
signer ; with it he can learn much about trans¬ 
ducer efficiency, and particularly about fre¬ 
quency dependence of efficiency. 

For a perfectly efficient transducer the con¬ 
stant-power response is a smooth curve, rising 
with frequency, following the behavior of the 
directivity index. Any peak or dip represents 
some form of anomalous behavior, perhaps dis¬ 
tortion of directivity patterns, and perhaps 
changes of efficiency. 

A perfectly efficient point source (i.e., direc¬ 
tivity factor equal 1) would produce 70.8 db 
above 1 dyne per sq cm for 1-w input. If a per¬ 
fectly efficient transducer has a directivity index 
of ~D db (Z) ^ 0) then, for 1-w input the pres¬ 
sure should be 70.8 -f D db in the direction of 
maximum response. If such a transducer had 
efficiency ~E db ^ 0) the pressure in the 
direction of maximum response should be 
70.8 -f D — £7 db. This relationship is a great 
help in estimating transducer efficiency; D can 
often be calculated within 1 or 2 db from the 
expression given in Section 4.4.2, and the effi¬ 
ciency so obtained is likely to be as accurate as 
can be obtained by present calibration methods. 

^ * * Constant-Available-Power 

Transmitter Response 

This transmitter response is one frequently 
encountered. In principle it is the response out 


of an idealized amplifier. The idealized amplifier 
consists of a constant emf in series with a con¬ 
stant resistance. The emf is adjusted to deliver 
1 w to a resistive load equal to the amplifier 
impedance. The resistive load is replaced by the 
actual transducer, and the response is reported 
as a function of frequency for constant amplifier 
emf. 

To the extent that this idealized amplifier 
simulates real amplifiers, the response curve 
resembles that to be expected in service. 

It has been common to report all kinds of 
transducers out of, say, 135-ohm or 600-ohm 
idealized amplifiers, regardless of the trans¬ 
ducer impedance. The reason for this was that 
such impedance taps are commonly available, 
and this reports the behavior to be expected 
when the transducer is so operated. Of course 
this is a defeatist attitude, implying that trans¬ 
formers are not to be had, and the apparent 
merit of a transducer so reported depends en¬ 
tirely on whether or not it happens to match 
the selected amplifier impedance. In order to 
judge properly the performance of any trans¬ 
ducer it is necessary to match the amplifier 
to the transducer. Furthermore, transducers 
should nearly always be tuned, usually with a 
coil at resonance, so that the ideal amplifier 
should be a complex-conjugate match to the 
transducer. Such a curve is then a very useful 
portrayal of the response to be expected when 
operated properly. 


1.6.0 Open-Circuit Receiver Voltage 

This is the open-circuit voltage, usually in db 
below 1 V, appearing across the transducer 
terminals, as a function of frequency, when a 
plane wave whose free-field pressure is 1 dyne 
per sq cm is incident upon the transducer. It is 
by far the most common way of reporting re¬ 
ceiver response, and is one of the most useful. 
Like the constant-current transmitter response, 
cable corrections must be made; these are tedi¬ 
ous and for small units may be in serious error. 
Also, like the constant-current transmitter re¬ 
sponse, it is independent of any series coils. 

Gain is cheap in receiver amplifiers, so that 
one is inclined to judge transducers on the basis 




STEADY-STATE RESPONSES 


167 


of open-circuit voltage regardless of impedance. 
From the standpoint of amplifier design this is 
nearly always justified, but for purposes of 
judging transducer performance it is not cor¬ 
rect. A fictitious improvement can be obtained 
by using thicker crystals or by connecting crys¬ 
tals in series, but the impedance changes so as 
to annul this apparent improvement. 

The open-circuit voltage reciprocates with 
the constant-current transmitter response, and 
so differs in shape only by a 6-db per octave 
slope. 

If care is taken to consider the shape of the 
array and the thickness of the crystals, open- 
circuit voltage offers a useful criterion of trans¬ 
ducer efficiency. The maximum value to be ex¬ 
pected for plane arrays is easily calculated, and 
failure to reach this is indicative of a commen¬ 
surate inefficiency; the method is so useful be¬ 
cause no account need be taken of the directivity 
index if the array is plane. 

These maximum values, predicted from 
Mason theory, for i/4-in. thick crystals, clamped 
drive, are: 

45° Y-cut RS: 62.4 db below 1 v, 

45° Z-cut ADP: 60.7 db below 1 v, 

for normally incident plane waves whose free- 
field pressure is 1 dyne per sq cm. 

The maximum occurs close to but not at the 
resonant frequency; for clamped Y cut it is 
slightly below resonance, for clamped Z cut 
slightly above. 


* Short-Circuit Receiver Current 

This response is not common but is coming 
into use because it does not require cable cor¬ 
rections. It is the current, usually in decibels 
below 1 amp, which flows in the short-circuited 
transducer terminals when there are incident 
upon the transducer plane waves whose free- 
field pressure is 1 dyne per sq cm. 

While advantageous because no cable correc¬ 
tion is required, this response is not much use 
to anyone. Transducer designers are accus¬ 
tomed to using open-circuit voltage and most 
numerical values are given in those terms. Elec¬ 
tronic engineers find the open-circuit voltage 
more to their liking. 


This response reciprocates with the constant- 
voltage transmitter, and differs in shape only 
by a 6 db per octave slope. It can, like open- 
circuit voltage, be used to estimate efficiency. 
The maximum values predicted by Mason theory 
are: 

45° Y-cut RS: 3.96 X amp, 

45° Z-cut ADP: 6.11 X amp, 

where n is the number of crystals all in parallel 
and all alike, and L„. is the width of a crystal in 
centimeters. Note that, unlike open-circuit volt¬ 
age, account must be taken of the number of 
crystals in the array. For this reason this is 
much less convenient and is not often used to 
estimate efficiency. 


Matched-Receiver Response 

This response is not now reported as calibra¬ 
tion data. It is the power, in db below 1 w, ex¬ 
pended in a complex conjugate load imposed on 
the transducer terminals when there are inci¬ 
dent on the transducer plane waves whose free- 
field pressure is 1 dyne per sq cm. Since trans¬ 
ducers are normally used near resonance, the 
load impedance should be chosen equal to the 
complex conjugate of the transducer’s imped¬ 
ance at resonance. It is difficult to obtain, and 
is of interest only to designers, but, like the con¬ 
stant-power transmitter response, is the only 
receiver response on which transducer perform¬ 
ance is properly judged. It should be peaked 
near the frequency of conjugate match, and 
should fall more or less symmetrically on either 
side. It offers a means of judging efficiency and 
frequency-dependence of efficiency, but is in¬ 
ferior to the constant-power transmitter re¬ 
sponse in this respect. 


System Response 

Nearly all transducers are intended for use in 
a sonar system in which they are driven by real 
amplifiers, often overloaded and through non¬ 
ideal transformers, frequently on long lengths 
of high-capacity cable, and often with equaliza¬ 
tion or with filters in the system. We cannot 
urge too strongly that it become common prac- 





168 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


tice to obtain transducer response data as op¬ 
erated by the actual system. Many times during 
World War II it was discovered that experi¬ 
mental systems behaved unexpectedly because 
of all the new conditions not included in cali¬ 
bration data. When calibrating system re¬ 
sponses a person familiar with the electronic 
equipment should be consulted to be sure that 
the data truly represent the behavior in service, 
particularly with regard to power levels. 


^ - TRANSIENT RESPONSES 

Many applications of transducers require the 
transmission of pulses 1 msec or less in time 
length. In particular are those applications re¬ 
quiring high repetition rates, which rates then 
are again dependent upon the pulse lengths. 
The shortest pulse that a transducer will handle 
is determined by its rise and decay characteris¬ 
tics, which are controlled by many factors, such 
as electrical Q, mechanical Q, or volume rever¬ 
beration inside the case and generally are func¬ 
tions of frequency. The prediction of transient 
behaviors from steady-state calibrations are 
thereby greatly complicated, and direct meas¬ 
urements of the acoustic pulse transduced are 
indicated. Using a probe pickup and pulse mod¬ 
ulator described in Chapter 9, pulses of 450- 
psec duration coming from a transducer whose 
resonance is 24.5 kc were recorded and are 
illustrated in Figure 22. The probe itself has 
a build-up time of 4 psec, which allows it to 
faithfully follow the pulses of Figure 22. 

Referring to the figure, the longest rises and 
decays appear to occur at the resonant frequen¬ 
cies. The decay tail seems to last about half the 
time of the pulse itself. At higher frequencies 
the tail seems to be shorter, but there seems to 
be a beat frequency between the applied and the 
resonant frequencies lasting over half the pulse 
duration. This characteristic appears at most 
frequencies, and is also found in the pulsing of 
electrical networks of lumped constants. 

There are many other applications of tran¬ 
sient testing, because this method facilitates the 
identification of particular acoustic paths by 
their lengths. In any underwater acoustic-test¬ 
ing system, unless the medium is infinite in ex¬ 


tent, reverberation due to reflection from the 
boundaries and other objects in the water are 
large and usually confuse steady-state measure¬ 
ments. With pulsing, however, effects from dif¬ 
ferent paths can be identified and each one 
measured separately. The kind of measure¬ 
ments benefited by this technique include cross¬ 
talk between two closely spaced units, direc¬ 
tivity patterns, free-field responses, etc. Figure 
22 illustrates a picture of the signal at the ter¬ 
minals of a receiving transducer whose radiat¬ 
ing plane is the same as that of a like trans¬ 
ducer being pulsed with 200-psec pulses and 
separated from the receiver by 6 in. The first 
large pulse is due to a direct path between the 
two units, and the remaining smaller ones are 
due to targets in the water 1- to 10-ft distant 
from the two units. The crosstalk from each 
path can thus be identified in both distance 
traveled and magnitude. 


4 8 LIMITLNG FACTORS 

In terms of a completed sonar system, the 
most important limitations of performance are 
the original specifications on which design was 
based. If a very small nondirectional unit is 
specified rather low-acoustic pressure must re¬ 
sult; allowing the physical size to increase al¬ 
lows a corresponding increase in output pres¬ 
sure. However, the questions of choice of design 
to meet specifications are discussed in Chapters 
6 and 7. In the present discussion the trans¬ 
ducer is regarded as a given thing, its design 
chosen for some unstated reason, and the in¬ 
herent limitations are considered. 


‘ ® ^ Steady-State Operation as a 
Transmitter 

Three possible limitations of output power in¬ 
herent to the transducer come to mind: current, 
voltage, and power input. Actually, the input 
current never imposes a limitation of itself, 
except possibly in series tuning coils which may 
saturate; such saturation is a trivial problem. 
If the transducer is operated at or very near 
resonance the limitation is nearly always caused 




LIMITING FACTORS 


169 



20 KC 



20 KC 

FORM OF INPUT PULSE 



30 KC 40 KC 



80 KC I I 0 KC 

Figure 22. Acoustic pulses of 450-nsec duration transmitted into water by the GD28-1 crystal transducer. 












170 


PROPERTIES OE ASSEMBLED CRYSTAL TRANSDUCERS 


by power; it is relatively easy to drive an effi¬ 
cient transducer to cavitation at resonance. 
This is discussed in Section 4.8.5. 

If the transducer is to operate over any ap¬ 
preciable frequency range, the maximum pos¬ 
sible power is likely to be a function of fre¬ 
quency, high near resonance and several db 
lower at the ends of the band. Near the ends of 
the band the limitation will be imposed by volt¬ 
age breakdown, usually across the edges of the 
crystals, but possibly even through the bodies 
of the crystals. 

The distance from resonance at which the 
limitation shifts from power to voltage is vari¬ 
able and indefinite. It will depend upon the de¬ 
tails of design (particularly on the geometry of 
the radiating face), upon the crystals used, and, 
above all, on the cleanliness observed in assem¬ 
bly. 

Thoroughly clean new Y-cut RS immersed in 
dry castor oil withstands up to 20 kv (60 cycle: 
rms) across thick crystals; at 20 kv 

nearly all samples break down. The breakdown 
of Z-cut ADP is not as definite, but under simi¬ 
lar conditions occurs at 30 to 40 kv (60 cycle: 
rms). In each case the breakdown is through 
the body of the crystal, and tends to initiate 
at a sharp electrode corner where the field 
strength is much greater than these numbers 
indicate. If rounded corners, etc., were used 
these crystals might stand several times as high 
voltage. 

Finished transducers rarely, if ever, achieve 
voltage limits as high as the new crystals. Dur¬ 
ing World War II, UCDWR was not able to 
maintain adequate standards of cleanliness, and 
the contamination of crystal edges during as¬ 
sembly reduced the voltage limit enormously. 
Representative UCDWR transducers using l^- 
in. thick crystals could not be relied upon above 
3,000 V if Y-cut RS, or 5,000 v if Z-cut ADP. 
The Bell Telephone Laboratories maintained ex¬ 
cellent cleanliness standards, and the corre¬ 
sponding limits on their units are much higher. 
No data are available on units made by Brush 
Development Company, Submarine Signal Com¬ 
pany, or Naval Research Laboratories [NRL], 
but these may well be higher than UCDWR. 

At resonance only about 1,000 v (rms) are 
needed across i/i-in. Z-cut ADP, or about 1,500 


V across Y-cut RS, to produce Ys w per sq cm 
if the crystal is clamped and fully loaded by the 
water. However this required voltage rises rap¬ 
idly as the frequency leaves resonance, and 
eventually a voltage limitation is met, usually 
only a small fraction of an octave from reso¬ 
nance. 


‘ ^ Short Pings as a Transmitter 

Since breakdown by power dissipation un¬ 
doubtedly involves local heating effects, it is 
plausible that the power limitation goes up for 
short pings and a low-duty cycle. If so, then 
even at resonance the ultimate limitation for 
sufficiently short pings would be voltage break¬ 
down through the crystal body. Under this con¬ 
dition there is no reason to anticipate input cur¬ 
rent limitations, particularly if air-core coils 
are used. 

If conditions were such that 20 kv could be 
put across i/4-in. 45° Y-cut RS, the resulting 
strain at resonance would be of the same order 
as the reputed maximum allowable strain for 
fracture (10^^) ; the maximum strain pre¬ 
dicted by Mason theory for 20 kv is approxi¬ 
mately 2.4 X 10~^. At resonance a maximum 
strain of 2.4 X 10~^ would radiate roughly 60 
w per sq cm, provided water, crystals, etc., re¬ 
mained linear. 

If it were possible to put 35 kv across i/4-in. 
45° Z-cut ADP crystals at resonance the inten¬ 
sity radiated into water would be roughly 430 
w per sq cm; no data are available, but this 
probably exceeds the maximum allowable strain 
for fracture. 


Partially Loaded Transmitters 

In the preceding sections it was assumed that 
the transducer was fully loaded by the radiation 
impedance of water. This is substantially true 
of crystals in a plane array whose dimensions 
are a couple of wavelengths or more, but it is 
usually not a good approximation for small ra¬ 
diating faces such as are often used to produce 
nondirectional patterns. Such faces may be 
loaded as little as 10 per cent of the full loading. 



LIMITING FACTORS 


171 


In such units the strain produced by a given 
voltage gradient may be many times those men¬ 
tioned in Section 4.8.2 and allowable strain 
might possibly impose a limitation before volt¬ 
age breakdown. 

The problem of cavitation (power) limits on 
such small radiating faces is understood quali¬ 
tatively, but any exact treatment requires a 
knowledge of the Green’s function from which 
the point impedance at each point in the radiat¬ 
ing surface may be calculated. Qualitatively, it 
is apparent that the loading will vary with 
position in the surface, and very probably is 
greatest at the geometric center. If the average 
radiation impedance is appreciably lower than 
the greatest point impedance, the average 
strain will be unduly great. The crystal face 
moves approximately as a plane piston, so the 
intensity radiated at the point of highest load¬ 
ing will be much greater than the average in¬ 
tensity. Thus, if such a poorly loaded face radi¬ 
ated an average intensity of only 0.1 w per sq 
cm it should not be surprising to find cavitation, 
since the intensity at some point might be 10 
w per sq cm. This behavior has been observed in 
several instances, notably UCDWR-type CY4 
(see Chapter 6) which shows signs of cavita¬ 
tion on the center line of each radiating face 
when driven above 1,500 v across i/4-in. 45° 
Y-cut RS. 


^ ‘ Receivers 

Except possibly in extremely quiet water it is 
not difficult to hear ambient water noise over 
ordinary transducers. Consequently a trans¬ 
ducer’s limitations as a transmitter are more 
important than the limitations as a receiver. 
However in some units, such as tiny probes, the 
self-noise inherent in the transducer may ob¬ 
scure ambient sounds. 

It is customary to think of the open-circuit 
voltage of a receiver worked into a grid circuit, 
without regard to its impedance. This attitude 
among system engineers is justified so long as 
the receiver is big. It is apparent that the quan¬ 
tity needed to evaluate correctly the sensitivity 
of a receiver is the power it will deliver to a per¬ 


fectly matched load impedance for unit incident 
sound pressure. As the area of the face of a re¬ 
ceiver is diminished this delivered power is 
diminished more or less proportionately, and 
unless the crystals are thinned the impedance 
goes up. Consequently a point is reached at 
which the “resistor noise” corresponding to this 
impedance obscures the sound coming from the 
water. As the accepted frequency band is in¬ 
creased this effect is made worse. Small “probe” 
units are often used as calibration standards, 
etc., in which service they are expected to be 
nondirectional at all frequencies and to have 
reasonably flat response over a wide frequency 
range. To achieve this the crystal is made very 
small and the resonance is placed well above the 
high end of the band. In such units all effects 
combine to raise the impedance (and inherent 
resistor noise) and lower the power delivery. If 
very low frequencies are expected of such very 
small probes, self-noise may limit performance 
in quiet water. Except for this rare require¬ 
ment, no inherent limitation is encountered 
down to ambient water noise. 


Cavitation 

The elementary classic theory of cavitation 
predicts that the liquid will rupture (cavitate) 
when the negative acoustic pressure equals am¬ 
bient. In water at zero depth this corresponds 
to a plane-wave intensity of Vs w per sq cm. Ex¬ 
tensions of this elementary picture take into ac¬ 
count surface tension and vapor pressure of the 
fluid. 

Most observers agree that castor oil and 
water do not cavitate at intensities a few db 
above Vs w per sq cm. The measurements are 
difficult and various observers disagree some¬ 
what on the exact intensity. There is good evi¬ 
dence to indicate that the presence of dissolved 
gas and microscopic bubble-forming nuclei in¬ 
fluence the observed cavitation intensity. 

The most remarkable feature of cavitation is 
the apparently random behavior under sup¬ 
posedly controlled conditions. One may set up 
cavitation in a beaker at controlled temperature 
(ambient) and after perhaps many minutes the 



172 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


cavitation may suddenly cease although the ap¬ 
plied voltage, etc., remain unchanged. As the 
voltage on a crystal is raised cavitation usually 
sets in at a definite voltage. If the voltage is 
then lowered the cavitation ceases at a lower 
voltage which appears to be a function of the 
time spent at cavitation. The voltage at which 
cavitation starts appears to be a function of the 
cavitation history of the liquid during several 
hours preceding. One usually associates with 
cavitation many different phenomena among 
which are: 

1. Bubble formation; this usually looks like 
a “smoke” in the liquid, but may contain large 
bubbles. 

2. A “frying noise” alleged to be caused by 
the collapse of millions of tiny bubbles. 

3. Nonlinearity of output acoustic pressure 
with input voltage. 

4. Great wave-form distortion of the sound 
in the liquid. 

5. Considerable temperature rise in the 
neighborhood. 

6. Pitting of the solid surface. 

Actually all of these are observed at one time 
or another, but they need not all occur simulta¬ 
neously. For example, one may set an ADP 
crystal cavitating in a beaker of oil, during 
which time bubbles leave the surface, and the 
wave form is seriously distorted (destroyed is a 
better description), yet sometimes no frying 
noise is heard, no great heating is observed, and 
no pitting results after many hours. On the 
other hand, a frying noise may be heard, the 
temperature may go up, and extensive pitting 
may occur with no visible bubbles. 

It is apparent that cavitation is not a single 
phenomenon, and one must define exactly the 
quantity he uses as a criterion for the occur¬ 
rence of cavitation. This is particularly impor¬ 
tant when comparing the results of two observ¬ 
ers, and some of the differences of opinion may 
result from use of different criteria. 

There is no doubt that the classic theory is in¬ 
adequate, and much interesting research re¬ 
mains before these phenomena are explained. 

For the purposes of this discussion of limi¬ 
tations of performance, the criterion for cavita¬ 
tion is taken as pitting or destruction of crystals 
after long-time operation. This is certainly the 


most fundamental limitation cavitation im¬ 
poses, although it is possible that wave-form 
distortion, etc., occurs at lower driving levels. 
Within this criterion, the following appear to be 
true: 

1. Under otherwise identical conditions ADP 
can radiate higher intensity than can RS. 

2. When fully-loaded plane arrays are used, 
both crystals can radiate more than Yg w per 
sq cm at zero submersion. 

3. The cavitation intensity rises with in¬ 
creasing depth of submersion. There is dis¬ 
agreement on how fast it rises or on the exist¬ 
ence of an upper limit. Probably a designer 
should not rely on this indefinitely until better 
data are available, and a 10-db increase is sug¬ 
gested as an upper limit (corresponding to 
roughly 300-ft depth). 

UCDWR has adopted the attitude that cavita¬ 
tion is a statistical affair; the power or voltage 
limit assigned by a designer is a function of the 
allowed risk on the particular transducer. If 
absolutely no risk is allowed the intensity at the 
crystal face should be held under Yi w per sq 
cm. If a very good risk is allowed, RS may be 
worked to Ys or even 1/2 w per sq cm, depending 
upon the geometry of the array, and ADP may 
be worked to 1 w per sq cm. If a definite gamble 
is desirable one may work RS to 1 w per sq cm 
and ADP to 3 w per sq cm. If, for some system 
development, it is desirable to radiate even 
higher intensities, tests should be conducted on 
the identical transducers to be used. All these 
figures are for zero submersion; appropriate 
increases for depth may be made. 

It is commonly assumed that transducers can¬ 
not or should not be driven above cavitation be¬ 
cause of destruction or because of loss of output 
power. For RS this is probably true since cavi¬ 
tation heating destroys the crystals very rap¬ 
idly. However ADP can tolerate very consider¬ 
able cavitation with minor or no damage. It is 
by no means certain that the output intensity 
does not continue to rise for increased electrical 
input above cavitation (albeit with a lower 
slope). If possible wave-form distortion may be 
tolerated to obtain higher intensity at the driv¬ 
ing frequency, it might well pay to operate above 
cavitation on some transducers. Extensive re¬ 
search on this subject is certainly indicated. 



ANALYSIS OF EQUIVALENT CIRCUITS 


173 


4 ^ ANALYSIS OF EQUIVALENT CIRCUITS 

At first sight the equivalent circuits for vari¬ 
ous transducers appear quite different both in 
magnitude and form, and it is not apparent 
what properties they have in common. Further¬ 
more, numerical calculation of impedance and 
response for each transducer is exceedingly 
tedious and to a large extent repetitive. It is the 
purpose of this section to restate the equivalent 
circuit in parametric form and to point out the 
tremendous number of similarities among 
transducers. From this parametrization expres¬ 
sions are derived which allow the compilation 
of a set of numerical data applicable to most 
transducers. 

For this purpose the Mason circuit is used 
without change; although this circuit embodies 
numerous approximations, no additional ap¬ 
proximations are made here. Thus these data 
present exactly the information given by 
Mason’s circuit. No correction is made for the 
finite width since actually all these results are 
width-dependent, but for ordinary purposes the 
changes are not important to preliminary de¬ 
sign, and are outweighed by the more serious 
couplings encountered in transducers. 

The predicted response curves, impedances, 
etc., are highly idealized. Some excellent trans¬ 
ducers depart considerably, but the majority of 
“ordinary” transducers agree with prediction 
rather well, provided they are efficient. These 
curves may be thought of as goals which trans¬ 
ducers approach and sometimes exceed. It must 
be emphasized that despite its limitations the 
Mason circuit does remarkably well for describ¬ 
ing transducer behavior, and in the present 
state of the art it is adequate for ordinary de¬ 
sign purposes. 


Mason Circuit 

The Mason circuit^ is given in Figure 23. The 
various quantities are: 

^ = “turns ratio” of an ideal electro¬ 
mechanical transformer. 


^ Some changes are made in the symbols for simpli¬ 
fication.la 


V = velocity of sound in the plated 
crystal in the direction, 

Zo = characteristic impedance of the 
plated crystal multiplied by the 
area L„.L<, 

Co = static electrical capacity of the 
crystal as a paraUel-plate con¬ 
denser with dielectric constant K, 
Ly, Lu; Lt = length, width, thickness. 

All quantities are given in cgs esu. Some 
relations among these quantities are: 


DKL^ 

^ = ~ ’ 

(69) 


(70) 

Zo = Lu-Li ^ p(l — k-)YQ, 

(71) 

^ ^ j/ 4.Y„’ 

(72) 

KL,..Ly 

(73) 

■ 47rL, ’ 


where D = piezoelectric constant, 

K = dielectric constant, 

Yo = Young’s modulus of the unplated 
crystal in the Ly direction, 
k = electromechanical coupling coeffi¬ 
cient. 

In order to convert mechanical resistance to 
electric ohms we divide all mechanical imped¬ 
ances by and remove the transformer (or 
change its ratio to 1/1). Having done this all 
impedances are in cgs esu. To convert to 
practical electrical ohms all impedances are 
multiplied by 9 X lO^b All dimensions remain in 
cgs units. 

This circuit, shown in Figure 24, is used for 
the balance of Section 4.9. 


Three Basic Drives 

Nearly all crystal transducers may be placed 
in one of three classes, depending upon the man¬ 
ner in which impedances are imposed on the two 
ends of the crystal. These are: 

1. Clamped drive: the radiation impedance 










174 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


(water) is on one end of the crystal, the other 
end is blocked by a backing plate. 

2. Symmetric drive: the radiation impedance 
(water) is on both ends of the crystal. 

3. Inertia drive: the radiation impedance 
(water) is on one end and zero impedance (air) 
is on the other. 

A given crystal operated in symmetric or in¬ 
ertia drive has the same resonant frequency as 
when free in air; operated in clamped drive the 
resonance is exactly half the free resonance 
(Mason approximation). 

Actually a backing plate presents an infinite 
impedance to a crystal at a single frequency 
(for wave plate) and then only if the plate 
is lossless, backed by vacuum, and attached by 


array so as to be fully loaded. This may not be 
true of small or curved units, or of arrays in 
which the crystals are spaced apart (see Sec¬ 
tion 4.2). 

In his book Mason considers only the clamped 
and inertia drives. For these two circuits he 
goes on to develop LC approximations to these 
trigonometric functions. To do this the LC 
circuit is chosen to resonate at the same fre¬ 
quency as the transcendental function’s first 
resonance, and the slopes at resonance are 
equated. This gives two equations for the two 
unknowns, L and C. However, the transcend¬ 
ental functions are multiperiodic and the LC 
circuits are not. Thus the approximation is 
good only near resonance; it remains good fur- 



Figure 23. Equivalent circuit of a transducer Figure 24. Equivalent circuit of a transducer 

with ideal transformer (Mason). with transformer removed or with an ideal 

transformer having all turns ratio. 


a perfect cement joint. However a sufficient ap¬ 
proximation for our present purpose is to con¬ 
sider the plate to present infinite impedance at 
all frequencies. 

The three basic circuits then became those 
shown in Figure 25. Trigonometric identities 
have been used to combine the cosecant and tan¬ 
gent terms. A transformation attributed to 
Norton^*^ has been used to obtain the inertia- 
drive circuit. 

In each case the radiation impedance imposed 
on a face of dimension is qiC]L„.L, mechan¬ 
ical ohms. To convert to electrical practical 
ohms this has been multiplied by 9X10^^ and 
divided by the transformer ratio. Here QtCi is 
the specific acoustic impedance of the medium 
(water) ; this tacitly assumes that the crystal 
under discussion is a member of a large plane 


ther below resonance than it does above, and is 
seriously in error at twice the resonant fre¬ 
quency. Mason’s results are showm in Figure 
26. The quantities shown have been converted 
to practical electrical units; Mason expresses 
them in cgs esu. 

In all cases these equivalent circuits repre¬ 
sent single crystals. If an array contains n crys¬ 
tals all in parallel and all identical, the circuit 
of the array is obtained from that of the single 
crystal by dividing every impedance by n. 


Parametrization 

We now define a set of parameters applicable 
to all the three basic circuits, and restate the 
three circuits in those terms. 































ANALYSIS OF EQUIVALENT CIRCUITS 


175 


Frequency 

In each case the first resonant frequency oc¬ 
curs when the series equivalent reactance of the 
mechanical branch vanishes. At this frequency 


which any other frequency (o is related to the 
resonant frequency: 

CO — a (On. 


. 2o«9«IO" , ujLy 

-I , - Cotan - - 

<1* V 






Co ^ 

5- i 


O 1 

■N 

►- 


-o-^ 


CLAMPED DRIVE 


. 2o»9«lO” , LjLy 


SYMMETRIC DRIVE 


/?, C^ L, Lt i9»IO" 


A C, L,,Lt.9.IO^ 

2^2 


Zo»9«IO*' _ u;L» 

—2--- Cotan - 

24i* 2V 



A C, Lw Lt«9«IO" 


Electrical Q 

In each case, at resonance, the equivalent 
circuit reduces to Co shunted by the radiation 
resistance. One might think of this as a “low- 
Q” condenser. Such a use of the symbol Q is 
common; the Q is defined as the shunt resist¬ 
ance divided by the shunt reactance. In this 
sense we define Qj, for a transducer: note that 
it is not to be regarded as a variable with fre- 


o 


Co ^ 


o- 




-)(—^ 


■o- 


Rm 


Co* 


4n Lt«9xl0” 


Farads 



Farads 


»9 xIO" 


Henries 


, x>i c,Lw Lt x9xl0” 


Ohms 


CLAMPED DRIVE 


Figure 25. Basic equivalent circuits of trans¬ 
ducers with transformers removed and with 
loads included in the circuit. 


placed across the radiation resistance. 
For clamped drive: 

cot ^ = 0, 


_ 

“ 2l; 


(74) 


0- i 

'——< 

r 

-c- 1 

C - 

Co ^ 

L c -L 

r ■ T 

CM 

O 

o 

Oi 

L* R ^ 

0-1 

•-4 

»- 

_ J 


Co* 

L, ■ 


XLwLy 


4^ Ltx9xl0*’ 
ZqL) 

2 Vlt) 


1-k* 


Co Farads 


x9xl0” Henries 
Ci 


c, L,, Lt X 9xl0<< 
4(tl* 


, ±1. 2L 
Cl L* ' 


For symmetric and inertia drives: 

(OliLy _ p. 

cot 2y' ~ 

ttV 

L’ 


(75) 


where is the resonant frequency in radians 
per second. 

We now define the frequency parameter a by 


INERTIA DRIVE 

Figure 26. Mason’s equivalent circuits for LC 
approximations to the trigonometric functions 
and with mechanical units expressed as prac¬ 
tical electrical units. 


quency, but as a parameter of the circuit, de¬ 
fined only at the resonant frequency. It is cus¬ 
tomary to ignore the algebraic sign of the re¬ 
actance, but for our purposes it is necessary 













































176 


PROPERTIES OE ASSEMBLED CRYSTAL TRANSDUCERS 


that it be retained; of any crystal is always a 
negative number. 

For clamped drive: 

^ PiCiLu'Lt X 9 X 10^'^ ^ 

~ -T; ‘ 

9" 

substituting" for (o^, </>-, and Co, the values given 
in Section 4.9.1 and above in Section 4.9.3, this 
expression reduces to 


approximation, and must define the mechanical 
by an extension of definition as follows. 

In a series LCR circuit the Q is defined: 



The series reactance of this circuit is 



U 

Qe = ■“ 27r“PiCi^2^ ■ 

For symmetric drive: 

^ PiC\LwLt X 9 X 10^^ ^ 

which reduces to 


(76) 


and at resonance 

corL = ^ 7 ^- 
wrL 

Now let us take the slope: 

^ _ r 4. J_ 

rfco “ ^ + co^C’ 


Qe — ~ 27r"piCi 


V 

D^K' 


(77) 


This is identical with clamped drive because, 
although the resistance is only half as great, 
the resonant frequency is twice as great and 
consequently the reactance is only half as great. 

For inertia drive: 


Qe 


piCiLwLt X 9 X 10^^ 

402 


<J^rCo, 


which reduces to: 


Qe 


^“PlCl^2X‘ 


(78) 


This is exactly one-half of clamped and sym¬ 
metric drive. 

Note that the crystal dimensions do not ap¬ 
pear in the expression for Qj^', it is a “crystal 
constant” whose value depends on the crystal 
material, the drive, and the water. 


Mechanical Q 

In the clamped and symmetric drives the me¬ 
chanical branch behaves near resonance like a 
series LCR circuit as shown in Figure 4. It is 
customary to speak of the Q of such a circuit. 
In this usage the phrase has a little different 
meaning from that used above in the electrical 
Q. In the series-resonant circuit the Q refers to 
the resonance and is defined as the inductive re¬ 
actance at resonance divided by the resistance. 
However, in this section we do not want the LC 


and evaluate it at resonance: 


Substituting this in the expression for Q : 



This becomes the definition of Like it is 
evaluated at resonance and is a parameter 
rather than a variable. 

For clamped drive the mechanical reactance 
is 


Zo X 9 X 10^1 o^Ly 
02 u * 

Differentiation by co generates a —csc^ which is 
unity at resonance, so 

/ ^ \ ^ Zo X 9 X 10“ ^ 

\ do3 j(i} = 0}R 0^ U 

Substituting in the expression for Q : 

^ ^ coi^02 Zo X 9 X 10^^ Ly 

2piCxL^Lt X 9 X 10^1 ’ 02U 

which may be reduced to 


Qm 


4 piCi 


(80) 


For symmetric drive we go through an analo¬ 
gous process and find that is identical with 
that of clamped drive. 

Inertia drive presents a more difficult prob- 










ANALYSIS OF EQUIVALENT CIRCUITS 


177 


lem. The presence of the tangent term shunting 
the radiation resistance complicates the expres¬ 
sions greatly. Upon differentiation one finds that 
the slope is indeterminate at resonance. Appli¬ 
cation of L’HopitaFs rule would be tedious and 
unpromising, so the slope is evaluated at co^ — e. 
The usual approximations are made for small e 
and then upon passing to the limit one obtains: 


dX \ ^ 9 X 10^^ 
dcx) 4</)- 


4ZoV[^^° (piCiL„,L,j J. 


To evaluate the series mechanical resist¬ 
ance at resonance is required. The shunting 
tangent term is therefore infinite so the resist¬ 
ance is just 


9 X 10^^ 
A4>- 


PiCiLwL t. 


Substituting in : 


2/2 W CO /co=a,« 

^jr/2pV_ pjC.\ 

4 \ piCi 2pV' 

This expression has an interesting form. The 
first term is just twice of clamped or sym¬ 
metric drive crystals. The second term arises 
because of the shunting tangent branch; since 
this branch is infinite at resonance it is not com¬ 
parable with the radiation resistance (at least 
for Y-cut RS or Z-cut ADP in water) anywhere 
near resonance. If one omitted the tangent 
branch the second term in would disappear 
and the value would be exactly twice that of 
clamped or symmetric. For 45° Y-cut RS or 45° 
Z-cut ADP the resulting error is negligible 
(about 3 per cent) but this approximation has 
not been made in the following sections. 

Notice that like is a crystal constant, 
dependent upon the crystal material, the radia¬ 
tion medium (water), and the type of drive, 
but not upon dimensions. 

Since the behavior with frequency is gov¬ 
erned entirely by and we see that all 
transducers containing a given kind of crystal 
in a given drive condition must be pretty much 
alike. The resonant frequencies may differ, but 
in octave measure this difference is removed, 
and the remaining differences are those of size 
alone. Of course this is a first approximation; 


we may expect various transducers to differ be¬ 
cause of second-order effects or because of re¬ 
duced efficiency. Examination will show that the 
behavior is a rather slow function of the extent 
to which the crystals are fully loaded, and not 
much difference is likely to come from this 
cause except through consequent changes of effi¬ 
ciency. The reason for this is that the QeQm 
product is the dominating term in calculating 
responses, and this product is independent of 

Qi<?i: 


QeQm — — 


il .pYi 

2 'D^K 



(82) 


Note that this product is the same as the ratio 
of capacities discussed by Mason^*^ and clearly 
indicates the fact that the achievable band 
width for any crystal depends wholly on k. It is 
remarkable that k for 45° Y-cut RS and 45° Z- 
cut ADP differ by only a few per cent; these 
crystals differ negligibly in achievable band 
width. At present there appears to be no hope 
for crystal transducers having broader band 
width except by the discovery of an otherwise 
suitable crystal having larger k. For example 
present band widths could be trebled by a ma¬ 
terial whose k equals 0.5 instead of 0.3. 


Resistance 

We now define the last parameter R as being 
equal to the radiation resistance. For generality 
w'e use the circuit for n crystals in parallel. 

For clamped drive: 

„ PiC.L^Lt X 9 X 10“ 

^ • 

For symmetric drive: 


p PiCiL^Lt X 9 X 10“ 
For inertia drive: 


R = 


PiCiLu-L t X 9 X 10“ 

4n(j)- 


(83) 


(84) 


Impedance 

In Section 4.9.3 it was shown that the two Q’s 
are crystal constants; from this we concluded 











178 


PROPERTIES OE ASSEMBLED CRYSTAL TRANSDUCERS 


that transducers containing the same crystals 
and drive may differ in magnitude, but not in 
the shape of their behavior with frequency. If 
so, we can obtain the complex impedance in 
parametric form so that a constant coefficient 
represents the size dependence and all the fre¬ 
quency dependence is contained in an expres¬ 
sion identical for all transducers with the same 
crystals and drive. 

For several reasons is a suitable quantity 
to act as the size-dependent coefficient; we seek 
an expression of the form 

Z = R{k,-^ jh). (85) 

Here Aq and k -2 will be functions of 
and a, but not of the number or size of the crys¬ 
tals. Then ki and k -2 may be calculated as func¬ 
tions of a for each kind of crystal in all three 
drives and the result allows the determination 
of the impedance of any transducer by simply 
evaluating R, a moment’s work. 

The task is simplified by the fact that and 
are identical for symmetric and clamped 
drive. Henceforth they are lumped together, 
and symmetric drive is always represented by 


2. For clamped drive. 


coLi, 


Zo X 9 X 10^^ 

ncf) 


air 

T’ 

-RQ.„ 

TT 


3. For inertia drive, 

CoL,j TT 

2V ~ "2’ 


Zo X 9 X 10^^ 
2«02 




Upon simplification it is found that R factors 
out of the expression for Z, leaving a complex 
number whose two components are the A*! and Aq 
sought. These are: 


For clamped drive 


kl — . 

a^QI + ( 1 - 

^cxQeQm cot Y ) 

(86) 

- aQs - cot 2 (1 


.(87) 

For inertia drived 




I 1 + 4 M-^ 


tan- 


k: 


.f].j4MHan=f + [2^ 

4:uQe cot- (air) 


air 

T 

xMQe 


„ ^ 2Mtan^ 

2cot^ - 2 _ 

tanf 




cot (avr) 


( 88 ) 


(89) 


clamped drive. In use however, symmetric drive 
requires its own value of R. 

To obtain expressions for k^ and Aq. requires 
much algebraic tedium not warranted here. In 
outline the method is: First obtain the expres¬ 
sion for the complex impedance seen looking in 
at the electric terminals. In this substitute the 
following expressions derived from the defini¬ 
tions of Qg and Q^i: 

1. Both clamped and inertia drive, 

_^ ^ 

TlcoCo OcQe 


These quantities are among those plotted as 
functions of a in Section 4.9.8. 


^ ^ Transmitter Responses 

Having obtained Aq and Aq. it is easy to calcu¬ 
late the power expended in the transducer for 
any selected method of operation. Since the 
equivalent circuit used assumes the transducer 

oT/ 

' For simplicity we define M = -. For 45° Y-eut 

Oici 

RS, M = 2.78. For 45° Z-cut ADP, M = 3.89. 


















ANALYSIS OF EQUIVALENT CIRCUITS 


179 


to be perfectly efficient this is the power radi¬ 
ated into the water. It is not possible to go on 
to give the acoustic pressure versus frequency 
unless the directivity index is known as a func¬ 
tion of frequency. This is discussed in earlier 
sections; here we stop with the power versus 
frequency, bearing in mind that for ordinary 
transducers the shape of the pressure response 
versus frequency will differ from the power 
curve only by a slope of 3 to 6 db per octave. 

Constant Voltage without Coils 

Transmitter calibrations of many trans¬ 
ducers are reported as constant-voltage re¬ 
sponse without tuning coils (see Section 4.6), 
so that the power expended in the transducer 
with E volts applied is of interest. This is 
simply 

The factor 

ki -\- ki 

is plotted against a in Section 4.9.8. 


Idealized Amplifier 

The available power response discussed in 
Section 4.6 has the advantage that it most 
nearly portrays the response to be expected 
when the transducer is driven by a real ampli¬ 
fier. The most useful curves are based on an 
idealized amplifier whose impedance is a conju¬ 
gate match to the transducer at resonance. In 
Figure H the minus sign on the coil reactance 


"SUf -iocRk2<q 



is necessary since ko is a negative number for 
any crystal whose reactance is capacitative at 
resonance. 

The power expended in the transducer when 
driven in this manner is 


Constant Voltage with Coils 

Some transducers have series tuning coils 
built into them. If coils are chosen to cancel the 
reactance at any frequency other than reso¬ 
nance the resulting response is so sharp that the 
transducer is virtually useless. For this reason 
we assume that the reactance is cancelled at 
a = 1. At a = 1 the reactance of the transducer 
is {Rk 2 )ct = \ so the reactance of the coil must 
be —iRk 2 )a = \- For simplicity we assume the 
coil to be lossless. Then the impedance of the 
transducer plus coil is 

R [ k\ + 7(^2 — ocik-i )]. 

When E volts are applied to the electric ter¬ 
minals the power expended is 


^ R'[h + + ife ~ 

The factor containing the k's is plotted in Sec¬ 
tion 4.9.8. 

Note carefully the similarity of this function 
for both Y-cut RS and Z-cut ADP, operated 
clamped (symmetric) or inertia. Although the 
other curves often look very different, this func¬ 
tion which portrays the response obtained in 
service is virtually identical for all six combina¬ 
tions of crystal and drive. This is a consequence 
of the dependence of the QeQ^[ product on k 
alone and of the fact that k is 0.3 for both 
crystals. 

Constant Current 


E^ _ h _ 

~R' ki+ [h - 


(91) 


The factor containing Zc’s is plotted against a 
in Section 4.9.8. 


The power expended in the transducer when 
a constant current of 1 amp is maintained is 
just Rki. Since ki is plotted in Section 4.9.8 for 
impedance, no constant-current response curve 
is shown. 
















180 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


Receiver Responses 

The short-circuit current receiver response 
reciprocates with the constant-voltage trans¬ 
mitter response and may be obtained in that 
way. Similarly the open-circuit receiver voltage 
reciprocates with the constant-current trans¬ 
mitter response and may be obtained from Rk-i_. 
Since the reciprocity merely introduces a slope 
of 6 db per octave, the receiver responses are 
not plotted in Section 4.9.8. 

Miscellaneous 

One or two other quantities of interest may 
be obtained from the equivalent circuit. Al¬ 
though some are best obtained without use of 
the parameters, they are given here to complete 
the section. 

Intensity 

Because of the limitations imposed on crys¬ 
tals by cavitation and voltage breakdown (see 
Section 4.8) it is important to have some esti¬ 
mate of the intensity radiated from the crystal 
faces at resonance when a given voltage is ap¬ 
plied. Unfortunately the highest point intensity 
depends upon the details of how the radiation 
load is imposed on each crystal face. However 
it is useful to have available the intensity which 
the Mason circuit predicts for crystals fully and 
uniformly loaded by the medium. For this we 
return to the equivalent circuits of Figure 3. 

In each case, Co plays no role. Furthermore, 
at resonance all the cotangent terms go to zero 
so the full voltage E is placed across the radia¬ 
tion resistance. If the power expended is divided 
by the radiating area we obtain the intensity 
of the radiation leaving the face in watts per 
sq cm. The results are: 

1. Clamped drive. 

Intensity = ( fJ lO-)' 

2. Symmetric drive. 

Intensity = (fJ{ui.ZfxloZ 

3. Inertia drive. 

Intensity = IQu )' (^ 5 ) 


The values of clamped and symmetric drive 
are the same because, although symmetric drive 
radiates twice as much power at a given voltage 
gradient, it does so from twice as great area 
(both ends of the crystal). The intensity radi¬ 
ated by inertia drive is four times that of 
clamped drive and thus offers a distinct advan¬ 
tage in situations where voltage breakdown 
may impose a limitation. 

Obviously the intensity is proportional to the 
square of the voltage gradient, as indicated. 


Peak Open-Cikcuit Voltage 

The effect on the equivalent circuit of an in¬ 
cident plane acoustic wave is to insert in series 
with the radiation resistance a zero-impedance 
voltage equal to the force exerted on the crystal. 
If the free-held acoustic pressure is p dynes per 
sq cm it is not known, in general, what this 
force is. For an inhnite plane array the force 
would be 2pLj^,Lf dynes (see Section 4.4.2) and 
that approximation will be used. In accordance 
with this we also assume the crystal to be fully 
loaded by the radiation medium. 

In order to express this receiver voltage in 
practical units it must be multiplied by 300 and 
divided by <t>. 

At resonance the cotangent terms go to zero 
and the tangent term to inhnity so the equiva¬ 
lent circuits become those shown in Figure 27. 
Symmetric drive is not included because in no 
practical case is it known how to treat the pres¬ 
sure imposed on the “back” end of the crystal. 

In each case the voltage is divided between Co 
and the radiation resistance, and the observed 
open-circuit voltage is that which occurs across 
Co. 

The algebra is straightforward; the result is: 


Peak open-circuit voltage 


=( 


2400. 


DK\/Ql + 


Z)PL,. 


(96) 


In obtaining this it was assumed that the 
maximum occurs at resonance. This is not ex¬ 
actly true; the analysis to determine the fre¬ 
quency at which it does occur is extremely tedi¬ 
ous, and the true frequency is quite close to 
resonance. Furthermore the true peak voltage 
differs little from that at resonance. 

Note that the peak voltage depends only on 








ANALYSIS OF EQUIVALENT CIRCUITS 


181 


the crystal thickness, not on the width, length, 
number, or arrangement. For this reason this is 
a very convenient measure of transducer be¬ 
havior. 

Peak Short-Circuit Current 

Using the same equivalent circuits (Figure 
27) we may calculate the peak short-circuit cur- 

o 


n Cq 


O- 

CLAMPED DRIVE 


^,c,LwLtx9KlO" 

n(J)* 



A c, LwLt«9«IO^' 

4n(J)* 


600 pL^Lt 

5 


Figure 27. Equivalent circuits of transducers 
operated as receivers at resonance. 


rent as a receiver for incident free field pres¬ 
sure p dynes per sq cm. 

When the electric terminals are shorted Co 
drops out of the picture; at resonance the series 
mechanical reactance vanishes, and the shunt¬ 
ing reactance of inertia drive is infinite. The 
radiation resistance alone remains, and the 
maximum short-circuit current is: 

Clamped drive, 


the thickness of the crystals, and depends on 
For this reason it is not so convenient for 
comparing transducers. 

Absolute Magnitude of Impedance 

In matching transducers to amplifiers it is 
important to have available the theoretical 
value of the absolute magnitude of the imped¬ 
ance. Since transducers are usually used with 
series tuning coils which cancel the reactance at 
the transducer’s resonance, such coils will be in¬ 
cluded in the calculation. 

The impedance of the transducer plus coils is 

Z = R\ki -\-j[k2 - 

hence, 

1Z| = RV + [h - (99) 

This quantity is plotted as a function of a in 
Section 4.9.8. 


Power Factor 


When matching an amplifier to a transducer 
the power factor is required as well as | Z [. As 
above, we compute the power factor with series 
tuning coils which cancel the reactance at the 
transducer’s resonance: 


PF = cos (j), 


cf) = tan“i 


I ^2 - j 


Thus, 


PF - 


k, 

V kiF [h - 


(lOOj 


This quantity is plotted as a function of a in 
Section 4.9.8. 


Numerical Values 


Inertia drive, 

Note that, contrary to open-circuit voltage, 
the peak short-circuit current is independent of 


In this section numerical values of these con¬ 
stants are tabulated for 45° Y-cut RS and 45° 
Z-cut ADP. Functions of a are plotted in the 
range 0.6 ^ a ^ 1.6; the curves may be read to 
the greatest justifiable accuracy. In all cases 
the numerical values are obtained from con¬ 
stants given by Mason. In some instances these 
constants were preliminary values, since 
























182 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 


changed by a few per cent, e.g., (1 — k-)Yo 
for Z-cut ADP is now taken to be 19.5 X 
but the curves have not been brought up to date. 
The constants used in these calculations are: 



45°Y-Cut RS 

45°Z-Cut ADP 

D 

11.2 X 104 

12.2 X 104 

K 

10.0 

14.2 

Yq 

10.8 X 1010 


- k-2)Yo 


18.9 X 1010 

Q 

1.775 

1.80 

k 

0.305 



(The difference in use of k arises from the 
manner in which these numbers were first re¬ 
ported to UCDWR; there is some doubt of the 
exact value of k for ADP but it is close to 0.30.) 

From these values the following quantities 
may be calculated, using the definitions given 
in earlier sections of Section 4.9. Any quantity 
dependent on qiCi is given for water as the radi¬ 
ation medium (piCi = 1.5 X 10-^). 


V 

Zo 

4 > 

Qe clamped 
Qe symmetric 
Qe inertia 
Qm clamped 
Qm symmetric 
Q.v inertia 
QeQm clamped 
QeQm symmetric 
QeQm inertia 

R clamped 
R symmetric 
R inertia 

Maximum intensity 
(resonance) 

Clamped 

Symmetric 

Inertia 

Maximum open-circuit 
voltage (reso¬ 
nance) (rms) 
Clamped 
Inertia 

Maximum short- 
circuit current 
(resonance) (rms) 
Clamped 
Inertia 


45°Y-Cut RS 

45°Z-Cut ADP 

2.35 

X 105 

3.24 

X 105 

4.17 

X lO^LuLt 

5.83 

X lO^Lu^Lt 

-8.91 

X 10^L,„ 

-13.8 

X 104L„ 

-5.55 


-4.54 


-5.55 


-4.54 


-2.78 


-2.27 


2.18 


3.05 


2.18 


3.05 


4.25 


6.01 


-12.1 


-13.8 


-12.1 


-13.8 


-11.8 

106L( 

-13.6 

7.14 

106L( 

17.0 


X 


106L( 

3.57 

106L< 

8.50 

X- r 

TXl-dw 

X nL. 

4.25 


1.79 

, 105L« 

X —r-^ 

X nLu, 

5.87 


14.0 


5.87 

xMf)’ 

14.0 



/ E \2 



23.5 

xMu) 

56.0 

xMu) 

1.18 

X 10-5 pLt 

0.938 

X 10-3 pLi 

2.41 

X 10-3 pLi 

1.90 

X 10-3 pLt 

3.96 

X 10-14 pnLr, 

6.12 

X 10-14 pnLu 

15.8 

X 10-14 pnL^ 

24.5 

X 10—14 pnLu 











0<7>OOK iD 


ANALYSIS OF EQUIVALENT CIRCUITS 



Figure 28. ki and ko as a function of a for clamped or symmetrically driven Z-cut ADP transducers. 
Z^Riki + jko). 



183 


































































































































































































































































































































































































































































































































































































































































184 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



Figure 29. ki and as a function of a for inertia driven Z-cut ADP transducers. Z — Riki + jk 2 ). 




























































































































































































































































































































































































































































































































































































































































00)00 <0 


ANALYSIS OF EQUIVALENT CIRCUITS 


185 



.0001 


Figure 30. /ci and ko as a function of a for clamped or symmetrically driven Y-cut Rochelle salt trans¬ 
ducers. Z = R{k\ + jko). 


































































































































































































































































































































































































































































































































































































































































A Ot ^OXOO 


186 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



.0001 


Figure 31. and Ao as a function of a for inertia driven Y-cut Rochelle salt transducers. Z — R{ki 
+ jAs). 















































































































































































































































































































































































































































ANALYSIS OF EQUIVALENT CIRCUITS 


187 

















































































































































































































































































































































































































































































































































































































































































































188 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



.6 .8 1.0 1.2 1.4 1.6 


a 

Figure 33. The quantity ki/kl + plotted against a in inertia driven Z-cut ADP transducers with 
constant voltage applied and without tuning coils. The ordinates of this curve when multiplied by E'^/R 
give the power expended in the transducer. 




































































































































































































































































































































































































































































































































































































































ANALYSIS OF EQUIVALENT CIRCUITS 


189 



Figure 34. The quantity ^i/^i + fii plotted against a in clamped or symmetrically driven Y-cut Rochelle 
salt transducers with constant voltage applied and without tuning coils. The ordinates of this curve when 
multiplied by E'^/R give the power expended in the transducer. 











































































































































































































































































































































































































































































































































































































































































































































































































































































190 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



Figure 35. The quantity plotted against a. in inertia driven Y-cut Rochelle salt transducers 

with constant voltage applied and without tuning voils. The ordinates of this curve when multiplied by 
E^/R give the power expended in the transducer. 



















































































































































































































































































































































































































































































































































































































































ANALYSIS OF EQUIVALENT CIRCUITS 


191 



Figure 36. The quantity +[^2 = plotted against a in clamped or symmetrically 

driven Z-cut ADP transducers with constant voltage applied and with lossless series tuning coil. The 
ordinates of this curve when multiplied by give the power expended in the transducer. 





















































































































































































































































































































































































































































































































































































































































192 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



.6 .8 1.0 1.2 1.4 1.6 

oc 


Figure 37. The quantity h\l}i\ + ~ <3!(^2)a = i]^ plotted against a in inertia driven Z-cut ADP 

transducers with constant voltage applied and with lossless series tuning coil. The ordinates of this curve 
when multiplied by /R give the power expended in the transducer. 















































































































































































































































































































































































































































































































































































































































ANALYSIS OF EQUIVALENT CIRCUITS 


193 



.6 .8 LO 1.2 1.4 1.6 

OC 


Figure 38. The quantity kilk\ + [^2 « (^2)0 = 1]^ plotted against a in clamped or symmetrically 

driven Y-cut Rochelle salt transducers with constant voltage applied and with lossless series tuning coil. 
The ordinates of this curve when multiplied by E'^/R give the power expended in the transducer. 









































































































































































































































































































































































































































194 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



.6 .8 1.0 1.2 1.4 1.6 

or 

Figure 39. The quantity k\/}i\ + [^2 ~ «(^2)a = i]^ plotted against a. in inertia driven Y-cut Rochelle 
salt transducers with constant voltage applied and with lossless series tuning coil. The ordinates of this 
curve when multiplied by /R give the power expended in the transducer. 



















































































































































































































































































































































































































































































195 


ANALYSIS OF EQUIVALENT CIRCUITS 



Figure 40. The quantity ki/[ki + {kx)a^i? + [^2 - « (^2)a = ]f plotted against a in clamped or sym¬ 
metrically driven Z-cut ADP transducers when connected to an idealized amphfier. (See Section 4.9.5.) 
The ordinates of this curve when multiplied by /R give the power expended in the transducer. 





































































































































































































































































































































































































































































































































































































































































































































































196 


PROPERTIES OE ASSEMBLED CRYSTAL TRANSDUCERS 



Figure 41. The quantity + (^i)a = i]" +[^2 —« (^ 2)0 = 1 ]^ plotted against a in inertia driven 

Z-cut ADP transducers when connected to an idealized amplifier. (See Section 4.9.5.) The ordinates of this 
curve when multiplied by give the power expended in the transducer. 



























































































































































































































































































































































































































































































































































































































































































































































ANALYSIS OF EQUIVALENT CIRCUITS 


197 



Figure 42. The quantity kil{kx + (^i)a = if + [^2 - « {k 2 )a = if plotted against a in clamped or sym¬ 
metrically driven Y-cut Rochelle salt transducers when connected to an idealized amplifier. (See Section 
4.9.5.) The ordinates of this curve when multiplied by E~/R give the power expended in the transducer. 



































































































































































































































































































































































































































































































































































































































































































































198 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



.6 .8 1.0 1.2 1.4 1.6 

CC 


Figure 43. The quantity ki/[ki + (^i)a = i]^ +[^2 —« (^ 2)0 = 1 ]" plotted against a. in inertia driven 
Y-cut Rochelle salt transducers when connected to an idealized amplifier. (See Section 4.9.5.) The ordi¬ 
nates of this curve when multiplied by /R give the power expended in the transducer. 









































































































































































































































































































































































































































































































































































































































































































































































































199 


ANALYSIS OF EQUIVALENT CIRCUITS 































































































































































































































































































































































































































































































































































































































































































































200 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



1 n il _ LJ 1 1. M ! ; I M 1 I I M 1 ! I.I I I I I I 

.6 .8 1.0 1.2 1.4 1.6 

or 


Figure 45. The data of Figure 41 plotted in terms of decibels below maximum power. 
































































































































































































































































































































































































































































































































































































































ANALYSIS OF EQUIVALENT CIRCUITS 


201 



Figure 46. The data of Figure 42 plotted in terms of decibels below maximum power, 









































































































































































































































































































































































































































































































































































































































































































































































































































































202 


PROPERTIES OE ASSEMBLED CRYSTAL TRANSDUCERS 

































































































































































































































1 







































































































































































































































































i 

































1 





























































] 







h 



1 















1 


















i 












A 






















4 

1 
























- 

























1 

































1 

















1 













1 




















1 













1 



1 










1 







1 
















1 

















1 
















It 





1 

































t 

























j 

1 

t 







I 



1 





















1 

1 
































1 

1 






J 



























Tn 





' 

i 


























Ml 


L 







I 




"i 

P 

1 






















_ 




j 

i 























; 

1 


1 






























hr 

“M 





































1 
































1 
































f 1 

1 
































j 




















































































































































1 

1 










































y 

































/ 
































i 



















1 














7 



















h 














f 
























































































































h 

r 
























































































































































n 





i' 


P 


























1 

in 


t' 














L 




I 













i 

I I 





1 














I 

















1 


1 


























H 

M 









r— 












1 











-J 

L 
































1 


































I 
























h' 









































r“ 

































I 

I 
































r ■ 

I 










































1 










p 























1_ 






























1“ 
















I 










r 


[1 

p 






























I 

















1 






i 






! 





I- . 


1 . 














1 



















t 














1 

















p 
















i 

1 

















L 
















P 




r 











I 


1 































I 


1 




















c 

























































i" 












1 









































I 























1 

































I 






















































r_; 

I 








1 

























I 

































r 























ll 










I 






















p 











L 






















I 









t_ 


t 

p 





















I 

1 











1 





















I 


j. 










i~ 































L 




















1~ 










ll 



1 


r 

























r 



L 






























r 

































1 


i_ 













INERTIA 45* Y CUT RS 


























1 

1 



1 


































































1 















































J 

















1 

































1 


! 














1 

1 
















1 


“t 














1 



















1 




1 








1 
















M“ 




J. 


L- 











1 











j 


p 



1 




1 













h 
















1 




i 





























1 




! 





























i 




! 















r 















“h 



; 

































"T 

' 


























































































1 
































I 



































1. j ' 

> j 






1 






















“I 



I 


11 

















L 


1 









'1 



I ! 






























■“1 


1 




1 1 


! 































\ ■ 


i 



























! 


1 




























‘t 


t" J 



























' ’ 1 




'll 































tA 1 




1 


























T\\ 














1 

i 












i 1 



1 . 1 



















i 









1 





V 


















“1 









P 

:j 





A 






















! 






u 

l'" 





■ 


























1 

I 






r; 


























_1 

I 

L_| 






I 


























1 

I 

! I 






1 


























I 

L, 






\ 


























I 








\i 


































































{ 

































\ 
































1 

\ 





































1 





















j 

1 










1 






















1 








1 































1 





























































I 


i 































1 


\ 



































































V 




















P 

r 












A 




















1-1 

1 —1 

h 

I 

i 











i 



















1 ^ 

I 


1 











\ 



















u 

































IJ 

I 














V 



















I 














A 


















1 

I 














\ 

1 

































v 




















i 













> 


































r 







1 












i 














\ 



















1 















i 







1 


1 
























V 









h 

p 

























































L 

















1 
















\ 





1 










i 















1 



M 
















I 














I 


1 
















1- 













L 




X 











































































































































































































































1 







1- 

































1 


























1 



























1 





















1 


















r 












1 
















1 





1 












































1 

































1 

































J 





























J_ 




J 





























1- 




L 
































1 




























' 




1 





1 









1 
















i- 



*1 




























r 

I 




p 

L 

j' 

1 









rj 














i 




t' 

‘T 



r 











p 










L 

j_ 

L 


“■ i 

±L 


J. 

_L 


L 


l; 



t 


Ll 














db 

0 

-2 

-4 

-6 

-8 

-10 

-12 

-14 

-16 

-18 

20 

•22 

•24 


.6 .8 

Figure 47. The data of Figure 43 plotted in terms 


1.0 I.: 

a. 


2 1.4 1.6 

of decibels below maximum power. 





























































































































































































































































































































































































































































































































































































ANALYSIS OF EQUIVALENT CIRCUITS 



Figure 48. Absolute magnitude of total impedance divided by R for a transducer whose reactance is 
cancelled at resonance by a lossless coil. This curve applies to clamped or symmetrically driven Z-cut ADP. 





























































































































































































































































































































































































































































































































































































































































































204 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



Figure 49. Absolute magnitude of total impedance divided by R for a transducer whose reactance is 
cancelled at resonance by a lossless coil. This curve applies to inertia driven Z-cut ADP. 










































































































































































































































































































































































































































































































































































































ANALYSIS OF EQUIVALENT CIRCUITS 


205 



1.0 1.2 
or 

Figure 50, Absolute magnitude of total impedance divided by R for a transducer whose reactance is 
cancelled at resonance by a lossless coil. This curve applies to clamped or symmetrically driven Y-cut 
Rochelle salt. 




















































































































































































































































































































































































































































































































































































































206 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



Figure 51. Absolute magnitude of total impedance divided by R for a transducer 'whose reactance is 
cancelled at resonance by a lossless coil. This curve applies to inertia driven Y-cut Rochelle salt. 




























































































































































































































































































































































































































































































































































































































ANALYSIS OF EQUIVALENT CIRCUITS 


207 



Figure 52. Power factor of clamped or symmetrically driven Z-cut ADP transducer with a lossless 
series coil which cancels the reactance at resonance. 





































































































































































































































































































































































































































































































































































































208 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



Figure 53. Power factor of inertia driven Z-cut ADP transducer with a lossless series coil which 
cancels the reactance at resonance. 









































































































































































































































































































































































































































































































































































































































































































































































ANALYSIS OF EQUIVALENT CIRCUITS 


209 



Figure 54. PowGr factor of clamped or symmetrically driven Y-cut Rochelle salt transducei with a 
lossless series coil which cancels the reactance at resonance. 






































































































































































































































































































































































































































































































































































































































































































































































































































































































































210 


PROPERTIES OF ASSEMBLED CRYSTAL TRANSDUCERS 



Figure 55. Power factor of inertia driven Y-cut Rochelle salt transducer with a lossless series coil 
which cancels the reactance at resonance. 














































































































































































































































































































































































































































































































































































































































































































































































Chapter 5 

ELECTRONIC SYSTEMS AND MATCHING NETWORKS 

By Francis X. Byrnes 


I N DESIGNING a Crystal transducer the design 
engineer should at all times keep in mind 
the fact that the transducer is only one link in 
a chain that makes up the complete electro¬ 
acoustic converting system. In any real system, 
with very few exceptions, the desired end result 
will be expressed as maximum power output for 
a given expenditure of weight and space, with 
the weight and space limitation applying to the 
system as a whole. With limitations of this type, 
the transducer designer should not make a 
change which raises the response of the trans¬ 
ducer by 2 db and adds 50 lb to its weight when 
this 50 lb of weight must be taken from the 
power amplifier and when the power amplifier 
in losing this 50 lb also loses 6 db in power 
output. 

Keeping in mind this intimate relationship 
that exists between the transducer and the 
other components in the complete electroacous¬ 
tic converting system, it is obvious that the 
transducer designer must concern himself with 
the other elements in the complete system. At 
the very minimum he must consider the cable 
used to connect the transducer to its associated 
electronic equipment. In addition to the cable 
he must also in the majority of cases concern 
himself with a matching network to raise the 
normally low power factor of the transducer. 
It may also be necessary to change the magni¬ 
tude of the impedance to a value that will match 
the associated amplifier. In the case of trans¬ 
mitting transducers in particular, the trans¬ 
ducer designer should concern himself with the 
properties of the associated amplifier. For ex¬ 
ample, he should take into account the manner 
in which the maximum power output of the 
amplifier varies as a function of the magnitude 
and power factor of the load impedance. 

In designing electroacoustic systems incor¬ 
porating crystal transducers for use over a 
band of frequencies, too often the entire burden 
of making the response flat over the band is 
thrust on the transducer with its associated 


coupling network. In many cases, and in par¬ 
ticular when the band width approaches an 
octave or more in width, the best overall effi¬ 
ciency is obtained by incorporating in the am¬ 
plifier an equalizing network to help flatten the 
response. This statement will usually apply to 
the efficiency regardless of definition, i.e., acous¬ 
tic watts out per electric watts in, watts per 
dollar cost, watts per pound, or watts per cubic 
foot of space occupied. The equalizer may be 
inserted in some portion of the electronic sys¬ 
tem several stages removed from the transducer 
terminals. 

While the transducer designer may feel that 
most of the foregoing design problems are more 
in the realm of the electronic design engineer, 
he will find that it will pay big dividends in im¬ 
proved performance if he will assume these 
problems himself, or at least cooperate closely 
with the electronic engineer in the work on 
these phases of the system design. 

The following sections will discuss these 
problems and will give suggested design pro¬ 
cedures with some practical examples. These 
procedures have been carefully worked out on 
paper but in most cases they have not been 
adequately tested to verify the predicted results. 
Therefore, they should not be taken as proven 
methods but should be considered as the best 
recommendations that the author is able to 
offer under the present state of transducer 
development. 


5 1 GENERAL PROPERTIES OE 

TRANSDUCERS 

Before going into a discussion of the other 
elements in the electroacoustic system, a brief 
summary of the general electrical and acoustical 
properties of crystal transducers is in order. 

Using the simple first-approximation equiva¬ 
lent circuit for a crystal transducer (Figure A), 


211 


212 


ELECTRONIC SYSTEMS AND MATCHING NETWORKS 


it can be shown'' that the curve of acoustical 
output as a function of frequency, of such a 
transducer for constant voltage or current 
applied to the electric terminals, will be a 
maximum at the resonant frequency of the 
mechanical system and will fall off as the fre¬ 
quency is removed from resonance. The rate at 
which the response drops off, i.e., the Q or 
sharpness of the resonance, depends on the Q of 
the mechanical system which in turn depends 
on the resistive component of the radiation im¬ 
pedance. This resistance, and therefore the 
sharpness of the response peak, can be con¬ 
trolled to some extent by various methods such 
as spacing the crystals in the array or by use of 
fronting plates. These methods of control are 
discussed in detail in Section 4.2. 

In an actual transducer, of course, there is 
more than one resonant system and, since most 
of these systems are made up of distributed 
rather than lumped elements, there will be 
many resonances associated with each system. 
However, in most cases there will be a well- 
defined resonance at the operating frequency 


Cm Cm 



or in the operating band, and all other reso¬ 
nances will either be so minor as to cause no 
trouble or will lie outside the operating band. 
The two notable exceptions to this general state¬ 
ment are: (1) the case in which a spurious 
resonance occurs (usually caused by coupling 
to the housing or to cavities around the crys¬ 
tals), and (2) the case in which the required 
frequency band is so wide that one or more of 
the crystal’s natural resonances are covered. 
The first of these exceptions may be remedied 

“ For a more detailed discussion of transducer analysis 
by means of the equivalent circuit see Section 4.9. 


by modification of the physical arrangement 
of the transducer. The second requires revision 
of the specifications to narrow the frequency 
band over which the transducer must operate 
or the inclusion of complicated equalizing net¬ 
works to compensate for the undulations of the 
response curve. 

A crystal transducer used as a receiver in a 
uniform sound field will have an output voltage 
which varies with frequency in the same gen¬ 
eral manner as does its output when used as a 
transmitter. This means that the same general 
problems are involved in equalizing the re¬ 
sponse of either receiving or transmitting 
transducers. 

Several examples of transmitter and receiver 
response curves are given in Figure 1, where 
the transducer output is plotted in decibels be¬ 
low an arbitrary zero reference level against 
the ratio a of the frequency under consideration 
to the frequency at which the open-circuit volt¬ 
age is maximum when the transducer is used as 
a receiver.'’ 

From the electrical standpoint a crystal 
transducer has an impedance essentially the 
same as a capacitor whose capacitance remains 
constant as a function of frequency except for 
variations near resonance which are usually 
small. The resistive component of this imped¬ 
ance varies considerably with frequency, rising 
to a peak very close to the resonant frequency 
and falling off on each side in a manner similar 
to the frequency-response curves. The relative 
magnitudes of the resistive and reactive com¬ 
ponents of the impedance are such as to cause 
the power-factor curve to have a typical maxi¬ 
mum value of from 0.2 to 0.3 at resonance, 
falling to values as low as 0.02 or 0.01 at fre¬ 
quencies off resonance. In some unusual cases 
the power factor will rise to values as high as 
0.7, and these transducers will show variations 
in capacitance of almost 3 to 1 when measured 
at frequencies near resonance. 

In designing matching networks for use with 
the transducer it is sometimes more convenient 
to refer to the electric Q of the transducer than 
to its power factor. This is defined as the 
ratio of the reactive to the resistive terms in 
its impedance at any given frequency. This 
See Chapter 4, Section 4.9.3. 












GENERAL PROPERTIES OF TRANSDUCERS 


213 


will range from 1 to 100 to correspond with the 
power factor range of 0.7 to 0.01 as given.® 

Figure 2 presents several examples of trans¬ 
ducer-impedance curves. Examples A, B, and C 
are curves applying to the same transducers 
whose response curves are shown in Figure 1. 

It should be noted here that transducers using 
X-cut Rochelle salt [RS] crystals will show 
much greater variations in both the resistive 
and reactive components of impedance as a 


RS and Z-cut ammonium clihydrogen phosphate 
[ADP] crystal transducers. 

One other property of a crystal transducer, 
which must be considered before going into the 
design of the accompanying electrical system, 
is its power-handling ability. For high-efficiency 
transducers used near resonance, which are 
operated continuously or employed to transmit 
pulses of more than 10 msec duration, the limi¬ 
tation will be cavitation in the liquid in contact 















/ 


"\ 





y 

r- 



\ 



/ 





\ 

\ 








\ 



.5 .6 .7 .8 .9 I. \2 1.5 17 2. 

a 


EXAMPLE C. XCCZ2-I 













f 

j 








/ 






/ 

/ 






/ 

/ 

/ 

/ 









.5 .6 .7 .8 .9 I. 1.2 15 L7 2. 


Figure 1. Examples of transducer response curves. The transmitter curves are taken with the applied 
voltage held constant. Those of the receiver I'epresent the open circuit voltage. All the curves have been 
reduced to zero (0.0) level at a = 1.0. 


function of frequency. The reactance curve may 
have regions in which it is inductive and will 
then pass through two or more points at which 
it is zero. However, since this impedance, as 
well as the resonant frequency, is a very marked 
function of both temperature and the applied 
field, it is impossible to treat quantitatively 
an X-cut RS transducer except under very 
carefully controlled conditions which are never 
encountered in the field. For this reason the 
following discussion will consider only Y-cut 

c Note that Qe is different from Qe defined in Sec¬ 
tion 4.9; at resonance, Qe = Qe. 


with the crystal face. The cavitation will dam¬ 
age the face of the crystal motor. If the effi¬ 
ciency of the transducer is low, if it is used far 
from resonance, or if it is to be used to transmit 
pulses much shorter than 10 msec the power 
limitation will usually be that at which voltage 
breakdown occurs in the crystals. 

The nature and power value of the limitation 
for any particular case cannot be stated accu¬ 
rately because of the many factors entering into 
the problem. However, the actual value of this 
limitation can always be stated in terms of a 
maximum voltage, current, or power, that may 























































































214 


ELECTRONIC SYSTEMS AND MATCHING NETWORKS 


be applied to the terminals of the transducer 
without damage to that unit. This electrical 
limitation should then be taken into consider¬ 
ation in the design of the driving amplifier so 
that its maximum output will not injure the 
transducer. 

A complete discussion of the power limita¬ 
tions of crystal transducers is given in Section 

4.8. 

CABLES 

In most applications of crystal transducers 
it will be necessary to use a length of cable 


true because of the fact that, if both cables have 
the same diameter, the capacitance between 
either conductor and the shield of a two-con¬ 
ductor cable is about the same as the capaci¬ 
tance between the single conductor and the 
shield in the single-conductor cable. Since the 
direct capacitance between leads in the two- 
conductor cable is very small compared to the 
capacitance to shield, the effective capacitance 
between conductors can be taken as the two 
conductor-to-shield capacitances in series. This 
is approximately one-half the capacitance of 
either conductor to shield. Thus if the shunt 
capacitance of a cable poses a problem, a two- 



Figure 2. Examples of the complex impedance curves of crystal transducers. The solid lines represent 
the resistance, the broken lines the reactance multiplied by (—1). All curve values have been multiplied 
by a factor which brings the reactance to 100 at a = 0.5. 


connecting the transducer to its associated 
electric equipment. In many cases the shunt 
capacitance of the cable will be so high as to 
exert an appreciable influence on the total im¬ 
pedance of the circuit. 

Only shielded cables having two conductors 
as used in a balanced line will be discussed. 
Such cables have a shunt capacitance that is 
approximately one-half that of a single-conduc¬ 
tor shielded cable of the same diameter, used in 
a single-wire line with shield return. This is 


wire balanced line will have a 2-to-I advantage 
over a single-wire unbalanced line. 

Commercially available two-conductor shield¬ 
ed cables for use in balanced lines will have 
values of effective lead to lead capacitance rang¬ 
ing from about 10 to 100 pqf per ft, with power 
factors ranging from 0.2 to 0.0005 over the 
frequency range of I to 150 kc. Figure 3 is a 
graph of capacitance and powder factor as a 
function of frequency for a widely used cable 
which has properties that are quite typical of 






























































































CABLES 


215 


the better synthetic rubber dielectric cables. 
Cables using polyethylene'^ as the dielectric have 
about one-half this capacitance and much lower 
power factors. The latter may be as low as 
0.0005. Cables using vinyl chloride or vinyl 
chloride acetate copolymers® as the dielectric 
will have somewhat higher capacitances than 
the rubber insulated cables and will have power 
factors as high as 0.2. 

In general, if a fairly good cable is used and 
it is not so long as to produce resonance effects, 


pacitance, will be a negligibly small fraction of 
the total power. 

The effect that the cable will have on the 
performance of the transducer will depend 
upon the manner in which its performance is 
being considered, the actual location of the 
cable in the system, and the value of the cable 
impedance relative to the other impedances in 
the system. The magnitude of the cable’s effect 
may be evaluated in several ways. The trans¬ 
ducer’s response may be measured with and 



Figure 3. Capacity per foot and power factor as a function of frequency for a 35-ft sample of Simplex 
No. 9061 (modified A AGO or SA60) 2-conductor shielded cable. 


the cable can be treated as a lossless capacitor 
shunted across the line. This treatment is al¬ 
most always valid because in any well-designed 
system the current flowing in the shunt capaci¬ 
tance of the cable will be a small fraction of the 
total current. Thus the power loss, represented 
by the 0.1 or smaller power factor in this ca- 

d Commercial designations of polyethylene plastics 
are Polythene and Copolene. 

e Commercial designations of vinyl chloride plastics 
are Geon and Koroseal; vinyl chloride acetate copol 5 nner 
is Vinylite. 


without the cable; or a standard current of 
1.0 amp may be applied through the cable and 
related to the resulting current in the trans¬ 
ducer. If the impedances of all the elements in 
the circuit are known the effect of the cable 
upon the transducer may be calculated. 

The following statements regarding the effect 
of the cable may be used as a general design 
guide. The extent to which the cable influences 
the response depends upon the ratio of the shunt 
impedance of the cable to the transducer im- 

















































216 


ELECTRONIC SYSTEMS AND MATCHING NETWORKS 


pedance, and also upon the value of the ter¬ 
minating impedance seen at the input end of 
the cable. When this impedance is zero (as with 
a constant-voltage source), the effect of the 
cable is also zero. In the more common case 
where a tuning coil is used to cancel the capaci¬ 
tive reactance of the transducer and the cable 
at some particular frequency, and this system 
is terminated by the proper impedance to give 
an efficient transfer of power at this frequency, 
the principal effect of the cable will be to cause 
a rapid fall of the response curve as the fre¬ 
quency of the applied voltage varies from the 
resonant frequency. It should be particularly 
noted that this sharpening of the electrical reso¬ 
nance curve is minimized if the tuning coil is 
placed between the cable and the transducer 
rather tlian at the amplifier end of the cable. 
This procedure places the cable in the low-im¬ 
pedance portion of the circuit and minimizes 
the effect of its shunt impedance. In the case 
where the transducer is used without a tuning 
coil the effect of the cable is to reduce the output 
level of the transducer without any appreciable 
effect on the shape of the response curve. In 
general it is seen that the lower the ratio of 
the cable’s shunt impedance to the impedances 
of the other elements in the circuit the greater 
will be the effect of the cable on the transducer’s 
performance. In extreme cases where a high- 
impedance transducer must be used with a long 
high-capacitance cable with a resulting low- 
shunt impedance, it may be necessary to use 
some kind of an impedance-matching trans¬ 
former between the transducer and the cable. 
If the transducer is used as a receiver the loss 
due to the cable may be greatly reduced by the 
use of a special input amplifier as described in 
Section 5.4. 

One extreme case is sometimes encountered 
where the required cable length is comparable 
to a quarter wavelength or more at the operat¬ 
ing frequency. For example, a 1,000-ft leng-th 
of the cable in Figure 3 becomes quarter wave 
at approximately 125 kc. The apparent discrep¬ 
ancy between this figure and that for a quarter 
wavelength in air at the same frequency is 
explained by the reduced velocity through cables 
of this type. The actual velocity will be that in 
air multiplied by a factor 1/k, where k is the 


dielectric constant of the cable insulation. In 
this case the dielectric constant of the rubber 
insulation was taken as 5.0. When the cable 
length is significant it can no longer be treated 
as a simple shunt capacitor. It must be treated 
as a transmission line with distributed con¬ 
stants and having a characteristic impedance 
which will be of the order of 100 ohms for all 
commonly used cables. In order to be used at 
the end of such a transmission line, a crystal 
transducer must have its reactance cancelled 
with a coil and the resulting resistive imped¬ 
ance transformed in magnitude to properly ter¬ 
minate the line, if losses due to reflections are 
to be avoided. Since a network consisting of a 
crystal transducer, its tuning coil, and an im¬ 
pedance transformer will in general appear 
resistive over only a comparatively narrow 
frequency range, such a system at the end of a 
long line will have a rather narrow response 
band. 


MATCHING NETWORKS 

Networks used for coupling crystal trans¬ 
ducers to electronic amplifiers may be resolved 
into two general types. The first, and simplest, 
of these has only to correct the power factor 
of the transducer. The second type, which is 
generally more complex, is used to transform 
its impedance to a higher or lower value. 

The design of a simple power factor correct¬ 
ing network is quite straightforward. The cir¬ 
cuit consists of an inductor in series or in paral¬ 
lel with the transducer. The reactance of the 
inductor is made equal to the reactance of the 
transducer at the frequency of operation. The 
coil must have a Q high enough to avoid serious 
power loss and must be insulated to withstand 
the applied voltage. It must also be capable of 
carrying the required current and in the case 
of iron-core coils the hysteresis loss must also 
be kept low. For the higher supersonic frequen¬ 
cies and for higher power levels it has been 
found that air-core coils can best meet the 
above requirements. For lower frequencies and 
for lower power levels the powdered iron or 
laminated iron core coils have been found supe¬ 
rior. For use in receiving transducers the best 



AMPLIFIERS 


217 


coil is one wound on a powdered iron toroidal 
form. By a proper choice of the core, very high 
Q’s may be obtained. The toroidal shape of the 
coil minimizes pickup due to stray electromag¬ 
netic fields. 

If a change in the magnitude of the imped¬ 
ance is desired in addition to correction of the 
power factor of the transducer, it will be nec¬ 
essary to have an impedance-transforming net¬ 
work in addition to the tuning coil. This may 
be a transformer of the required turns ratio or 
it may be a simple L- or T-type impedance¬ 
transforming network. If in designing the im¬ 
pedance-matching transformer the leakage re¬ 
actance is deliberately made high and is ad¬ 
justed to the proper value, it may also serve as 
the required tuning inductance. Special care 
must be taken, however, to insure low losses 
in this leakage inductance. In most cases, an 
L network consisting of a condenser and a coil 
can quite satisfactorily carry out the functions 
of both the tuning coil and an impedance-match¬ 
ing network. Sometimes special circumstances 
will make it desirable to use other types of 
networks. An example of this is the case in 
which it is desirable to obtain a large impedance 
transformation. A simple shunt coil will in 
general be better than the alternate L network 
for the reason that less inductance is required 
enabling a smaller coil with lower losses. Since 
the coil is the principal factor determining size, 
cost, and losses, in a reactive network, the one 
using the smaller coil will be the best, assuming 
these three factors alone are considered. 

If the system is to be used over a band of 
frequencies it will be found that the network 
having the least narrowing effect on the pass 
band will always be the one employing the least 
reactive elements, namely, a simple series or 
sometimes parallel coil whose reactance is equal 
to the reactance of the transducer at the mid¬ 
band frequency. If an impedance transforma¬ 
tion must be made in such a band-pass coupling 
system it can be made with the least effect on 
the band width by means of a transformer 
whose response in the pass band is flat. Such a 
transformer is almost always feasible to build 
by conventional methods because of the fact 
that the maximum pass band likely to be en¬ 
countered in this application is about 1 octave. 


" ‘ AMPLIFIERS 

The most flexible link in the chain that makes 
up the complete electroacoustic system is the 
electronic amplifier that either supplies the 
driving power for the transducer or is driven 
by the transducer. This great flexibility is due 
to the fact that all the design factors, including 
physical shape, output power, input impedance, 
output impedance, and frequency response are 
more or less independently variable over a very 
wide range of values. These same factors which 
enable flexibility in the design also make it 
very difficult to establish quantitative design 
rules. This is particularly true in the case of 
an output-power amplifier which is operated 
very near its maximum output. In this case the 
independence of some of the design factors is 
to a great extent lost, and the rather high 
degree of nonlinearity makes necessary the in¬ 
troduction of many higher order terms in any 
mathematical treatment. The particular load 
impedance that we are here concerned with, a 
crystal transducer, adds even more complica¬ 
tions because of the wide Variation in magni¬ 
tude of its impedance as a function of frequency 
and the fact that its power factor will have a 
low average unless the operating frequency 
range is restricted to a narrow band near reso¬ 
nance. 

For most practical design purposes, however, 
the following qualitative rules will be found to 
give satisfactory results. 

We will first consider transmitting transduc¬ 
ers operating at a single frequency or over a 
very narrow band of frequencies. The reactance 
of the transducer plus the connecting cable and 
any impedance transformer should be cancelled 
by means of a series tuning coil. The magnitude 
of the resulting purely resistive load should then 
be made equal to that value which will permit 
delivery of maximum power, at the allowable 
distortion, from the particular tubes being used 
in the output amplifier. The actual value of this 
optimum-load impedance may be obtained from 
examples given in a standard tube handbook 
or it may be obtained by analytic methods^ from 
curves given in the handbook. 

The second and most important case to be 
considered is where the system is to operate 




218 


ELECTRONIC SYSTEMS AND MATCHING NETWORKS 


over a fairly wide band; that is from 0.1 to 1 
octave in width. The resonant frequency of the 
transducer should occur at the geometric center 
(defined as the frequency ^/fjo) of the band. 
A tuning coil, that will cancel the reactance of 
the transducer and its associated coupling net¬ 
work at this frequency, should be used. The 
impedance of such a combination will show 
variations in magnitude as large as 10 to 1 and 
even greater variations in power factor over 
such band widths. In Figure 4 are shown the 


is used, a set of equal power contours plotted on 
a plane of impedance coordinates can be ob¬ 
tained.- These contour maps can use for imped¬ 
ance coordinates X and R, \Z\, and Q^, or \Z\ 
and power factor [PF]. The particular choice 
will depend on the form in which the impedance 
data have been presented. All three have been 
plotted in Figures 5, 6, and 7. The values of 
X, R, and \Z\ are given relative to R^, in the 
equivalent circuit. This equivalent circuit and 
the power-impedance contours derived from it 


PF |z| 




.5 .7 .8 .9 I L2 1.5 1.7 2 

Ot 



.5 .6 .7 .8 .9 I 1.2 L9 1.7 2 


Figure 4. Examples of the impedances of transducers which include a lossless series tuning coil such 
that the combination resonates as a = 1.0. Solid line curves are absolute magnitude of impedance, broken 
line curves are power factor. 


impedances of the three transducers given as 
examples in Section 5.1. They have here been re- 
plotted in terms of magnitude of impedance 
and power factor and include the effect of a 
series tuning coil resonating at the midband 
frequency. 

In order to decide how the magnitude of such 
a load should be adjusted relative to the value 
that the output amplifier would like to see, we 
will have to take into consideration the effects 
that varying load magnitude and power factor 
have on the maximum output capabilities of 
the amplifier. If the conventional first-approxi¬ 
mation equivalent circuit^-'* for a vacuum tube 


will be found to be quite accurate for push-pull 
class Ai triodes, using R^ equal to the plate 
resistance of the tube, if fairly high distortion 
can be tolerated. (With transducer loads this 
high distortion is eliminated to a large extent 
by the filtering action of the transducer and 
its associated coupling network.) For push-pull 
class Ai beam tetrodes, or pentodes, the 
method will still provide fairly accurate results 
if a value of R^^ is made equal to the optimum 
value of load, rather than the actual plate re¬ 
sistance. For these tubes the error arising from 
the use of these curves is much larger if the 
load is greater in magnitude than it is if the 



































































































AMPLIFIERS 


219 


load is smaller in magnitude than the optimum 
value. This large error for high-load impedances 
is not a problem as it will be found that in 
almost every case wherein the load is a crystal 
transducer, the amplifier will not be called upon 
to operate into a load of higher than optimum 
impedance. The XCCU6Z transducer given as 
an example above, when matched in a manner 


pentode amplifiers if they are not operated too 
close to the overload point, and if the load-power 
factor remains sufficiently high. If more accu¬ 
rate performance predictions must be made for 
these tubes operated under these conditions, a 
special set of power-impedance contours must 
be obtained experimentally for the particular 
circuit being used. 



Figure 5. Curves showing the effect of varying the load impedance R ± jX of an ideal amplifier having 
an internal impedance of 1.0 + jO ohms. The power level contours are labeled in decibels below the 
maximum power capability of the amplifier when operating into a resistive load of 1.0 ohms. The reactive 
component (Z) of the impedance is plotted as the ordinate and the resistive component (R) as the 
abscissa. The curve X = R is plotted for convenience and its use is described in the text. 


to give maximum band width, is an exception 
to this rule. However, because the power factor 
is very good and the excitation to the amplifier 
(if it is properly equalized) will be reduced at 
the frequencies where the impedance is above 
the optimum value, the operation will still be 
satisfactory. 

These curves will also apply fairly well for 
single-ended class Ai triode, beam tetrode, or 


Class ABi, ABo, and Bo amplifiers depart even 
farther from the simple basis from which the 
above curves were derived. The class ABo and 
Bo amplifiers are complicated by the fact that 
the grids will require some driving power from 
the preceding stage and the amount of power 
required will depend both on the driving voltage 
and upon the load in the plate circuit of the 
driven tubes. When this load varies in the 









































































220 


ELECTRONIC SYSTEMS AND MATCHING NETWORKS 


manner of a crystal transducer it is almost 
impossible to establish general rules for the 
results that will be obtained under various con¬ 
ditions. Another factor complicating the class 
ABi, AB 2 , and B 2 amplifier design is that the 
power drawn from the power supply increases 
with grid excitation, and if the load has a low 
power factor, as do transducers in parts of the 


pation is not to be exceeded. For the shorter 
pulses the thermal inertia of the anodes of the 
tubes can help to reduce the effect of this un¬ 
usually high anode heat, and this factor will be 
much less important. 

While the power-impedance contours for the 
class ABi, AB 2 , and B 2 amplifiers will have the 
same general form as those for the ideal ampli- 



Figure 6. Curves showing the effect of varying the load impedance of an ideal amplifier having an 
internal impedance of 1.0 + jO ohms. The power level contours are labeled in decibels below the maximum 
power capability of the amplifier when operating into a resistive load of 1.0 ohms. The Q of the load 
impedance is plotted as the ordinate and the absolute magnitude (|Z|) of the impedance as the abscissa. 


band, it will not absorb much of this increased 
power input, and the excess will have to be dis¬ 
sipated as heat at the anodes of the tubes. For 
steady-state conditions, or for comparatively 
long pulses, this will mean that the maximum 
power output of the amplifier will have to be 
reduced if the maximum allowable plate dissi- 


fier, they will probably differ sufficiently to 
make it advisable to prepare a special set from 
experimental data taken on the particular tubes 
that are to be used. 

If impedances of the type given in Figure 4 
are plotted on the same coordinate paper as 
Figure 7 (see Figure 8 for example) the re- 














































































AMPLIFIERS 


221 


suiting plot can be superimposed on Figure 7, 
and by sliding the two plots relative to each 
other, keeping the abscissas in alignment, the 
effect of various impedance-matching ratios can 
be tried. On the plots using X and R as imped¬ 
ance coordinates the plots should be slid along 
the diagonal line so that the X = R lines are 


width at the 10-db down points at the expense 
of the midband frequencies then the transducer 
impedance plot should be slid down until it cuts 
the R^ = 1 line at the —10-db contour. (See 
solid curve. Figure 9.) This makes the frequen¬ 
cies of the — 10-db points a = 0.5 and a = 1.7, 
and the midband response has been dropped to 



Figure 7. Curves showing the effect of varying the load impedance of an ideal amplifier having an 
internal impedance of 1.0 + jO ohms. The power level contours are labeled in decibels below the maximum 
power capability of the amplifier when operating into a resistive load of 1.0 ohms. The power factor (PF) 
of the load impedance is plotted as the ordinate and the absolute magnitude (|Z|) of the impedance as the 
abscissa. 


kept superimposed. Taking, for example, the 
XCCZ2-1 and putting its impedance plot on the 
power-impedance plot in such a position as to 
make its impedance at midband fall on the 
Rp = l line (see broken-line curve on Figure 9), 
it is seen that the frequencies of the 3-db down 
points are a = 0.85 and a = 1.2, and the 10-db 
down points fall at a = 0.7 and a = 1.5 approxi¬ 
mately. If it is desired to improve the band 


nearly —4 db with the former —3-db frequen¬ 
cies, a = 0.85 and a = 1.2, being now nearly 
—5 db in level. 

Looking at the solid curve of Figure 9 and 
noting that the value of transducer impedance 
where it crosses the Rp = 1 line is equal to 58 
ohms, it is seen that to achieve the indicated 
results the amplifier used will have to have a 
value of Rp equal to 58 ohms or it must be pro- 





































































222 


ELECTRONIC SYSTEMS AND MATCHING NETWORKS 


vided with an output transformer which will 
transform the actual of the amplifier to this 
value. An alternative method would be to adjust 
the thickness of the crystals in the transducer 
until its impedance was of the proper value to 
match the particular amplifier. 

It should be noted that in achieving the proper 
match, there are three elements that may be 


the cable between the amplifier and the trans¬ 
ducer is so short that its shunt impedance is 
high as compared with the amplifier-output im¬ 
pedance, it is often quite feasible to make the 
impedance of the transducer match the ampli¬ 
fier directly (perhaps by connecting crystals in 
series), thus eliminating the output-matching 
transformer. 



Figure 8. Impedance of the XCCZ2-1 crystal transducer plotted on jZj and PF coordinates The value 
of a IS shown at points along the curve. . c vdiuo 


varied if the whole system is being designed 
as a unit: the impedance of the transducer, the 
impedance ratio of the matching network, and 
the output impedance of the amplifier. By keep¬ 
ing this in mind when the system is being de¬ 
signed, it will often be found that a simple 
modification of one element may result in a 
major simplification of one of the other ele¬ 
ments. For example, if the system is such that 


In using these graphs it should be kept in 
mind that they show only the amount by which 
the power delivered to the transducer is reduced 
by mismatching. The actual sound intensity 
that the transducer delivers in a given direction 
is also affected by its directivity and efficiency, 
both of which are functions of frequency. If, 
for example, the transducer being matched is a 
highly directional unit and the desired end re- 


































































AMPLIFIERS 


223 


suit is a flat pressure versus frequency response 
in the direction of maximum intensity then it 
will be necessary to compensate for the approx¬ 
imately 6-db per octave rise in output caused by 
the increase in directivity with frequency. In 
order to reduce this effect it will usually be 
found helpful to make the frequency at which 
the tuning coil and transducer resonate a little 


ance contours, and sliding them up and down 
until the parts of the desired band that are 
lowest in response have the best possible match. 

Even after the impedance match has been ad¬ 
justed to give the most uniform response curve 
that is possible it will often be found that the 
results are not good enough. In this case an 
equalizer, as discussed in Section 5.5, should be 



Figure 9. Examples of two different conditions of impedance matching for XCCZ2-1 transducer. 


lower than the frequency at which the trans¬ 
ducer is mechanically resonant. If, on the other 
hand, there is some source of inefficiency pres¬ 
ent that increases with frequency it may even 
be desirable to place this electrical resonance at 
some frequency higher than that of mechanical 
resonance. In any case the effect of changing 
the value of the tuning coil can be observed by 
plotting the impedance with different coils, 
superimposing these plots on the power-imped- 


used to produce the final flattening of the re¬ 
sponse curve. 

It should be noted that a class C amplifier 
may also be used resulting in very high effi¬ 
ciency at a single frequency when the electrical 
Q of the transducer plus its coupling network 
is high enough to reduce the attendant distor¬ 
tion to a reasonable value. The design of a class 
C amplifier for this service is quite conven¬ 
tional. 



























































224 


ELECTRONIC SYSTEMS AND MATCHING NETWORKS 


It is worthy of note that many applications 
of crystal transducers use such short pulses that 
pulsed power amplifiers, as used in radar, may 
be utilized. The use of such amplifiers will per¬ 
mit smaller tubes for a given output power, or 
much more power from a given tube. So far as 
the transducer is concerned, a pulsed power 
amplifier will act the same as an equivalent 
class B or C amplifier capable of continuous 
operation. Since this is true, all the above state¬ 
ments regarding class B and C amplifiers, ex¬ 
cept those that apply to plate dissipation prob¬ 
lems, will also apply to pulsed power amplifiers. 

In considering receiving transducers and am¬ 
plifiers for use with them, the design situation 
is found to be much simpler than the problems 
involving transmitting transducers. There are 
only two conditions that must be satisfied in the 
receiving system. First, the impedance match 
should be such that the output power of the 
transducer is coupled efficiently to the amplifier. 
Second, the noise voltage in the amplifier should 
be less than the noise voltage in the transducer 
output due to acoustic noise in the medium. 

This second condition is very easily satisfied 
under almost all conditions because of the high 
transducing efficiency of crystal transducers 
and because of the high ambient-noise level that 
is encountered under even the quietest condi¬ 
tions of the sea. The one condition under which 
the amplifier self-noise is likely to be the limit¬ 
ing factor is that in which the pass band is in 
the extremely low-frequency end of the audio 
spectrum, and a small low-capacitance hydro¬ 
phone is being used. Under these conditions the 
impedance of the hydrophone is very high mak¬ 
ing it extremely difficult to obtain an efficient 
transfer of power to the amplifier. This tend¬ 
ency towards lowered output under these spe¬ 
cial conditions, emphasizes the importance of 
the best possible impedance match between 
transducer and amplifier. Special precautions 
should be taken to keep the amplifier self-noise 
at a low level. 

The first condition is satisfied by a procedure 
essentially the same as used with transmitting 
transducers. That is, for single-frequency oper¬ 
ation the mechanical resonance of the trans¬ 
ducer should be at the frequency of operation 
and a tuning coil should be used to cancel the 


reactance of the transducer at that frequency. 
The resulting resistive impedance should then 
be transformed, if necessary, to match the input 
impedance of the amplifier. For wide band op¬ 
eration the resonance of the transducer should 
be at the geometric center of the band and a 
tuning coil should again be used to cancel react¬ 
ance of the transducer at this frequency. 

The power-impedance contours given in Fig¬ 
ures 5, 6, and 7 may be used in adjusting the 
impedance match in exactly the same manner 
as with transmitting transducers. The value of 
Rp on the contour will in this case represent the 
input resistance of the amplifier. Since the 
power levels in receiving circuits are alw^ays 
quite low, the input resistance of the amplifier 
will be a constant, and the contours may be 
used without the reservations necessary in the 
case of power amplifiers driving transmitting 
transducers. 

Because of the low-power levels encountered 
in receiving systems there is a special input 
amplifier that may be used to greatly reduce 
the effect of cable capacitance shunted across 
the transducer. In order to use this amplifier the 
connecting cable must either be a single-wire 
unbalanced line, or if it is a two-wire balanced 
line each wire must be in its own separate 
shield. The shield is then driven with respect 
to ground by an amplifier which applies to it 
a voltage which, as nearly as possible, should 
be the equal in both phase and amplitude of the 
voltage on the signal wire. This gives an effec¬ 
tive guard action which will completely elim¬ 
inate the elTect of the signal lead-to-shield ca¬ 
pacitance, provided the applied voltage is 
exactly equal to the signal voltage. If these 
voltages are not equal the reduction factor is 
equal to 

Signal voltage or A + 1 

Signal voltage-guard voltage ’ 

where A is the gain within the feedback loop. 

Practical circuits for accomplishing this re¬ 
sult are given in Figure 10. The first type of 
guard amplifier, using only one cathode-loaded 
stage, will be useful to provide a capacitance 
reduction factor of the order of 10, with guard- 
shield-to-ground shunt impedances of the order 
of 1,000 ohms, if a tube similar to a 6AC7 is 





AMPLIFIERS 


225 


used. The second type of guard amplifier using 
two amplifier stages plus a coupling stage will 
give much higher values for the reduction fac¬ 
tor and will handle much higher values of 
guard-shield-to-ground shunt capacitances. The 
actual values of input-capacitance reduction 


shield-to-ground capacitance of 5,000 ppf and 
a high-frequency cutoff of 200 kc. If the value 
of the feedback resistor is increased by a 
factor of 10 the input capacitance-reduction 
factor will be increased to 1,000 but the maxi¬ 
mum value of guard-shield-to-ground capaci- 



A = gain measured from the grid of the first tube to point ® with the negative return of the first stage made to ground rather 
than to point ®. The values of C, and should be chosen to give the required low-frequency response. The value of Cj^ should 
be chosen so that its capacitance in parallel with the input and output capacitances of the associated tubes and the stray 
wiring capacitances will have a shunt reactance equal to Rj^ at 200 kc. All resistor values are in ohms. 

Figure 10. Circuits for guard amplifiers. 


factors that can be obtained and the actual 
values of guard-shield-to-ground capacitances 
that can be tolerated, are dependent on each 
other and on the high-frequency cutoff of the 
system. With the circuit constants as given in 
Figure 10 the input-capacitance reduction fac¬ 
tor will be approximately 100 with a guard- 


tance that can be tolerated will be only 50 pqf, 
if the 200 kc high-frequency cutoff is to be re¬ 
tained. If the high-frequency cutoff is lowered 
by a factor of 10 the tolerable guard-shield-to- 
ground capacitance is increased by a factor 
of 10. 

In designing a three-stage guard amplifier it 
































































226 


ELECTRONIC SYSTEMS AND MATCHING NETWORKS 


should be kept in mind that it is essentially a 
three-stage feedback amplifier with a very large 
amount of feedback. In order to avoid insta¬ 
bility the gain and phase-shift characteristics 
of the amplifier, with the actual input and out¬ 
put loads connected, must be controlled very 
carefully up to extremely high frequencies. In 
some cases this control must be exercised to 
1.000 times the highest frequency to be ampli¬ 
fied. If additional phase-shift and gain-control 
networks are included in the amplifier, it is 
possible to obtain results somewhat better than 
those attributed to the simple amplifier. For 
example, the amount of guard-shield-to-ground 
capacitance that can be tolerated may be in¬ 
creased by a factor of about 3. 

For design information for feedback ampli¬ 
fiers, and phase-shift and gain-control networks 
for use with them, refer to the bibliography. 

In applying these circuits to two-wire lines 
the required separate shield on each wire neces¬ 
sitates the use of separate guard amplifiers for 
each wire. 

In order to minimize the possibility of stray 
pickup of interference in the ground system the 
cable used in this circuit should have an overall 
shield which is insulated both from the guard 
shield and from ground, except at one point. 

This guard circuit finds particular applica¬ 
tion in the fairly common case of a small low- 
capacitance hydrophone to be used at the end 
of a long cable and at low frequencies. In con¬ 
nection with this application it should be noted 
that when the a-c return of the input-grid re¬ 
sistor is made to the guard-voltage circuit, as 
shown in Figure 10, its effective value as seen 
by the transducer will be increased by the same 
factor that the cable capacitance is decreased. 

The actual voltage gain between input and 
output terminals of the guard amplifier will be 
approximately 1 in the case of the single-stage 
amplifier and 200 in the case of the three-stage 
amplifier. 


5 5 EQUALIZING NETWORKS 

In some applications of crystal transducers 
the flatness of the pass band that can be ob¬ 


tained with the aid of a simple coupling network 
between the transducer and its associated am¬ 
plifier is not satisfactory. In this case the fre¬ 
quency response of the amplifier will have to 
be modified in such a manner as to give the 
overall response of the amplifier-coupling-net- 
work-transducer combination the desired flat¬ 
ness over the required band. 

In theory at least, an equalizer may be built 
to modify the amplifier response in any desired 
manner. In actual practice it will be found that 
virtually all good transducers, when coupled 
through the correct tuning coil to an amplifier 
having an impedance which gives the most effi¬ 
cient transfer of power over the band, will have 
a response curve showing a more or less flat- 
topped peak which will fall off with fair uni¬ 
formity on both sides of resonance. (Correct 
tuning coil in this case is the one which cancels 
the reactance of the transducer at its frequency 
of mechanical resonance.) Such a response 
curve may be flattened with an equalizer con¬ 
sisting of a single ji or T-section band-pass filter 
which is terminated by impedances that are 
higher than the design loads. A band-pass filter 
used in this manner gives the necessary flat- 
bottomed valley in the middle of the pass band 
with the response curve rising to peaks at the 
edges of the pass band. An actual example of 
the results obtained by using such a filter to 
equalize a system consisting of a receiving 
transducer, amplifier, output tuning coil, and 
transmitting transducer is shown in Figure 11. 
The shape of the response curve of this type 
of equalizer may be varied over a wide range of 
width of pass band and height and sharpness 
of the peaks at the edges of the pass band. 

A design guide for this type of equalizer with 
the circuits used and response curves obtained 
is given at the end of this chapter. 

If the response to be equalized is anything 
other than a simple resonance peak some other 
type of equalizer must be used. An excellent 
discussion of equalizers of various shapes is 
given in reference 3. 

Equalizer Design Guide 

The following design procedure for this type 
of equalizer has been found to be satisfactory 



EQUALIZING NETWORKS 


227 


when the usual space limitations are imposed. 
In the following discussion it will be assumed 
that the entire filter is to be small enough to 
fit into a can 4 in. high and 2 in. square. 

The first step is to choose the nominal imped¬ 
ance of the filter. This should be as high as pos¬ 
sible so as to realize the maximum possible gain 
from the amplifier stage which uses the filter 
as its plate load. In practice it will be found 
that the highest impedance value that can be 


and 40 kc the coil that will best meet these re¬ 
quirements is a toroid with a powdered Permal¬ 
loy ore. At higher frequencies a solenoid-type 
coil with a powdered iron core will be best. 

The tuning capacitor C used in the jc-type 
filter should be chosen so as to resonate with 
the coil L at the upper edge of the pass band, 
and it should resonate at the lower edge of the 
pass band in the case of the T-type filter. The 
coupling capacitor should then be adjusted 



10 15 20 25 30 35 40 

RECEIVER RESPONSE INTO AMPLIFIER 



10 15 20 25 30 35 40 


TRANSMITTER RESPONSE OUT OF 
AMPLIFIER WITH OUTPUT TUNING 
COIL 



10 15 2 0 25 30 35 40 

COMBINED RESPONSE OF 
RECEIVER AND TRANSMITTER 



Figure 11. Example of results obtained by equalizing a system consisting of a receiving transducer, 
amplifiers, output tuning coil, and transmitting transducer over the band of 14 to 28 kilocycles. Frequency 
in kilocycles is plotted in the abscissa. 


obtained will be about 10,000 ohms and that the 
limiting factor will be the coils. The procedure 
then, is to find the highest inductance coil that 
will fit into the available space and will meet 
the electrical requirements. The Q must be 
greater than 10 and the natural resonance of 
the coil must be at least one octave higher than 
the upper cutoff frequency of the filter. It will 
be found that at frequencies lying between 1 


to locate properly the lower edge of the pass 
band, in the case of the n-type filter, or the 
upper edge of the pass band, in the case of the 
T-type filter. This last adjustment should be 
made with the R/s having a value of at least 
100,000 ohms and with shorted out. 

The value of the R^’s and R^ should then be 
adjusted to give the desired shape of response 
using Figures 12D, E, and F as a guide. It 







































































RESPONSE 


228 


ELECTRONIC SYSTEMS AND MATCHING NETWORKS 



A. It type filter 

Cb SHOULD BE AT 





AMPUFIER CONNECTIONS TO FILTER 



D. CURVES SHOWING EFFECT OF VARYING 
RpIN EITHER FILTER WITH Rs=0 




VARYING Rs IN XT TYPE FILTER VARYING Rs IN T TYPE FILTER 

Figure 12. Equalizer design guide. 
















































EQUALIZING NETWORKS 


229 


should be noted that if it is desired to have the 
response higher at the high-frequency end of 
the band, the Jt-type filter will have to be used, 
while if the low-frequency end is to be higher 
the T-type filter must be used. It should be kept 


in mind when adjusting the R’s, that the RJs 
affect the height of the peaks on both ends of 
the band, while Rg affects the height of the lower 
peak only in the case of the ji-type filter, and the 
upper peak only in the case of the T-type filter. 





Chapter 6 

DESIGN PROCEDURES 

By T. Finley Burke 


T his chapter is concerned with the elemen¬ 
tary procedures involved in designing a 
transducer to meet given specifications. Our 
attitude is optimistic; the Mason circuit and its 
simple consequences are used, little attempt 
being made to anticipate the innumerable ways 
in which a transducer may misbehave. The de¬ 
sign selected on this basis is by no means com¬ 
plete, requiring the modifications discussed in 
Chapter 7. However, all the major features are 
usually settled by this simple approach, and the 
preliminary design will look and act much like 
the final unit if design is properly carried out. 

‘'i SPECIFICATIONS 

The need of a complete set of performance 
specifications before design can start is obvious. 
These should result from a conference between 
the consumer and the designer. Specifications 
drawn up without such a conference are likely 
to request impossibilities or else to request less 
than optimum performance. In drafting the 
specifications the transducer should be regarded 
as one component of an integrated system, and 
consideration should be given whether a par¬ 
ticular feature (e.g., flat frequency response) 
is best accomplished in the transducer or else¬ 
where in the system. Furthermore, some esti¬ 
mate should be made of the volume of future 
production in order to weigh the advantages of 
novel or expensive materials. 

The following list of items which may require 
specification is more detailed than is usually 
required in any one instance. It is intended as a 
check list for suggestions. 

1. Tactical use of the complete system includ¬ 
ing mode of operation. 

2. Transducer to be used as transmitter, re¬ 
ceiver or both. 

3. Frequency response. 

a. Number of db down from maximum at 
stated frequencies. 


b. Allowable local variations in frequency 
response. 

c. Overall flatness required; allowable 
slope. 

d. These properties to obtain when driven 
from or working into specified amplifier 
in specified manner. 

4. Directivity patterns at specified frequen¬ 
cies. 

a. Width of main lobe in horizontal and 
vertical planes. 

b. Lobe suppression in horizontal and ver¬ 
tical planes. 

c. Axial or other symmetry required. 

d. Directivity index. 

e. Extent of suppression of backward ra¬ 
diation. 

f. Permissible departures from ideal. 

g. Special features, such as bearing devi¬ 
ation indicator [BDI]. 

5. Power levels at frequency and direction of 
maximum response. 

a. Pressure in the water at stated distance 
from transmitter. 

b. Ping length and duty cycle of transmit¬ 
ter. 

c. Voltage to be delivered across specified 
impedance by receiver for unit incident 
sound pressure. 

d. Required minimum efficiency. 

6. Complex impedance as a function of fre¬ 
quency. 

a. Approximate magnitude required. 

b. Tuning coils or other matching net¬ 
works. 

c. Ratio of series reactance to series re¬ 
sistance : 

(1) Frequency of minimum ratio. 

(2) Required minimum, allowed maxi¬ 
mum. 

(3) Specified frequency dependence. 

d. Nature of amplifier to be used. 

e. Length and nature of electric cables, 
voltage restrictions. 


230 


CHOICE OF CRYSTAL MATERIAL 


231 


f. Other electrical components. 

7. Special features required. 

a. Horizontal or vertical BDI. 

b. Maintenance of close contact [MCC]. 

c. Phase steering of directivity patterns. 

d. Low crosstalk between transmitter and 
adjacent receiver. 

e. Other special features such as intention 
to operate simultaneously at two fre¬ 
quencies. 

8, Service conditions. 

a. Maximum and minimum ocean depth to 
which transducer will be subjected. 

b. Maximum speed at which transducer 
will be moved through water. 

c. Whether transducer will be rotated; 
streamlining to be provided. 

d. Maximum accelerations to be encoun¬ 
tered ; dropping, depth charge, gun 
blast. 

e. Temperature conditions to be encoun¬ 
tered. 

(1) In shipment and storage. 

(2) In operation. 

f. Any especially rough handling antici¬ 
pated. 

g. Weight, size, shape, and density require¬ 
ments. 

h. Corrosion problem to be expected. 

(1) Frequency of inspection, painting, 
and repair. 

i. Any special provisions for repair in the 
field, 

j. Service life, whether or not expendable 
after single use. 

Crosstalk is best specified at a particular frequency 
as follows. Let the transmitter be driven in some con¬ 
stant (unspecified) manner so as to produce an 
acoustic pressure P at unit distance in the direction 
of maximum intensity. When the transmitter is driven 
in this manner, the crosstalk causes an electric signal 
in the receiver. If this signal in the receiver were 
caused by a normally incident plane acoustic wave, the 
pressure of this hypothetical wave could be deter¬ 
mined from the frequency response calibration of the 
receiver. Then the crosstalk is defined: 

Crosstalk (in db) = 10 log • 

This definition is concerned with an operationally 
useful quantity; it avoids distinctions between electric 
and acoustical couplings, difficulties in the method of 
measurement, and the particular character of the as¬ 
sociated electric circuits. 


k. Other components of the system, such 
as a dome, which directly influence de¬ 
sign. 

l. Mounting position on ship or other 
body; possible interferences from ship’s 
hull, shroud lines, cavitation bubbles 
from other structures, ship’s wake, etc. 

m. Required mounting holes or studs, gas¬ 
kets, packing glands, terminal strips, 
access holes, nameplates, camouflage 
paint. 

9. Manufacturing requirements. 

a. Allowable cost. 

b. Expected volume of production. 

(1) Immediate. 

(2) Pilot models. 

(3) Large production. 

c. Time allowed. 

d. Any special manufacturing facilities 
available. 

e. Should manufacturer’s tests be devised? 

(1) On components and subassemblies. 

(2) Bench tests on full assembly. 

(3) Water tests. 

f. Any considerations of packaging for 
shipment and storage. 

g. Any requirements for operating and 
service manuals to be prepared. 


^2 CHOICE OF CRYSTAL MATERIAL 

To all intents this book is limited to 45° Y-cut 
Rochelle salt [RS] and 45° Z-cut ammonium di¬ 
hydrogen phosphate [ADP] ; little is given here 
or anywhere else concerning 45° X-cut RS and 
its design parameters. In certain applications, 
notably in small probe units (e.g.. Brush De¬ 
velopment Company Model C-11), X-cut RS has 
the advantage of low impedance without serious 
limitations caused by temperature and field de¬ 
pendence. In those applications, operating far 
below resonance, the information given by 
Mason in his book^ is adequate for design. The 
X-cut RS near its resonance is properly the 
subject of another book the size of this one, but 
it is the concensus that goffers no advantages 
to outweigh the temperature and field de¬ 
pendence. 

In choosing between 45° Y-cut RS and 45° 




232 


DESIGN PROCEDURES 


Z-CLit ADP the designer must have in mind all 
the different factors discussed in this and the 
next chapter. However, some generalizations 
may be made. 

On the whole Z-cut ADP is preferable to Y- 
cut RS because of the greater range of tempera¬ 
ture over which it is safe and because of the lack 
of water of crystallization. Probably as a conse¬ 
quence of the foregoing advantages, Z-cut ADP 
can radiate significantly higher intensity with¬ 
out cavitation damage (see Section 4.8). Be¬ 
cause of the higher temperatures allowed, better 
cement joints can be made to Z-cut ADP; in 
particular, the Cycle-Weld technique for attach¬ 
ing crystals to rubber windows requires ADP 
(see Section 8.5.6). 

It will be seen from Section 4.9.8 that al¬ 
though the constant-current and constant-volt- 
age response curves differ, there is little to 
choose between when driven out of an amplifier. 
ADP leads to lower-impedance transducers for 
a given resonant frequency, a small advantage 
in its favor. However crystals about IV 2 times 
as long for a given frequency are required if Z- 
cut ADP is used instead of Y-cut RS; at low fre¬ 
quency such long crystals may not be available. 
Also, ADP has much lower volume resistivity 
than does RS, and in transducers to be operated 
far below resonance this might limit perform¬ 
ance. Lastly, the “vertical-stack” design em¬ 
bodied in University of California Division of 
War Research [UCDWR] Model CY4 (see Sec¬ 
tion 6.8) is very useful for certain applications, 
but to date no method has been found for using 
this design with Z-cut ADP. 

The choice is summed up by saying that Z-cut 
ADP is preferable unless some particular rea¬ 
son (such as low frequency, or vertical-stack 
design) indicates otherwise. 


CHOICE OF BASIC DESIGN 

Having chosen the crystal material, the de¬ 
signer must next select the basic design: 
clamped, symmetric, or inertia drive (see Sec¬ 
tion 4.9). 

If symmetric drive is desired it is probably 
for a more or less nondirectional vertical stack, 
and Y-cut RS is required. The only other com¬ 


mon use of symmetric drive is in small probe 
units (e.g.. Brush Development Company C-11) 
in which symmetric drive is easiest and most 
natural in order to produce the nondirectional 
pattern usually required. 

Because the uses to which symmetric drive is 
put are so specialized, no competition exists be¬ 
tween it and the other two designs. Where sym¬ 
metric drive is good it is so much better, and 
where it is poor it is so much worse, that the 
choice is clearly marked. 

Symmetric drive is unexcelled for line sources 
(vertical stack) which are to have moderate 
directivity in one plane and to be more or less 
nondirectional at right angles. However such 
vertical stacks have an inherent power limita¬ 
tion dependent upon the radiating area; if the 
area is insufficient there is nothing to be done 
but abandon the design in favor of a cylindrical 
unit using inertia or clamped drive. 

In almost every case inertia and clamped 
drives compete. The selection depends upon the 
details of the problem and upon personal opin¬ 
ion. The pros and cons may be summarized. 

1. Clamped drive advantages: 

a. Oldest style; best understood. 

b. Inherently rugged. 

c. Naval Research Laboratory [NRL] 
unit construction eliminates many 
troubles from backing-plate vibrations. 

d. Relatively high radiation impedance in 
equivalent circuit diminishes effect of 
a given loss resistance. 

e. Oil allows high voltages. 

2. Clamped drive disadvantages: 

a. Usually efficiency is at least 2 or 3 db 
below 100 per cent. Probably viscous 
losses in oil and cement cause this. 

b. Heavy. 

c. Relatively higher voltage required for 
given power output. 

d. If large backing plates or oil cavities 
are used, spurious vibrations may harm 
patterns or response. 

3. Gas-filled inertia drive advantages; 

a. Glued joints are not at point of high 
stress, thus give less trouble. 

b. Patterns usually excellent. 

c. No oil to give viscous losses. 



SIZE OF CRYSTAL 


233 


d. Efficiency usually very high. 

e. Performance highly predictable. 

f. Relatively lower voltage for given power 
output. 

4. Gas-tilled inertia drive disadvantages: 

a. Difficult to strengthen window in large 
units. 

b. Relatively new; Cycle-Weld cement not 
tested over many years. 

c. Relatively low radiation resistance in 
equivalent circuit makes a given loss 
more important. 

d. Special filling with Freon (or similar) 
gas required to allow high voltage. 

Oil-filled inertia drive is subject to the worst 
troubles with clamped drive. These are viscous 
losses, spurious modes, etc. On the other hand 
it has few of the advantages of gas-filled inertia 
drive. 

The curves showing response when driven by 
an ideal amplifier (see Section 4.9.8) indicate 
that there is little reason to choose one design 
over the other on the basis of Mason theory; 
the choice must be based on the features inher¬ 
ent in the present methods of construction, and 
is subject to change as improved designs appear 
and as more is learned of present design. 


^ SIZE OF CRYSTAL 

The next design step is the selection of the 
size of crystal. This is rather simple. It is al¬ 
ways desirable to place the resonance of the 
transducer at the geometric center of the speci¬ 
fied operating band of frequencies. If the crys¬ 
tal is to be used in an inertia drive the dimen¬ 
sions are obtained from equation (62), Section 
3.2.3. The only limitation is that it is well to 
keep the width-length ratio less than 1/2 and 
the thickness-width ratio less than 1/2. 

The only uses to which symmetric drive is 
put are such that the radiation reactance lowers 
the resonance significantly; for example, in 
UCDWR Model CY4 the observed resonance is 
22 V 2 kc in water although the resonance in air 
is 27.6 kc. The uncorrected size for a given fre¬ 
quency is the same as the inertia-drive size for 


that frequency; this may then be corrected for 
the radiation reactance expected. 

If a crystal is to be mounted on a clamping 
backing plate (i.e., wave thick at resonance), 
the backing plate should be regarded as a mir¬ 
ror in which the crystal sees its own image. The 
resonant frequency is then that predicted by 
equation (62) of Section 3.2.3 for a crystal as 
big as the clamped crystal plus its image. For 
infinitely slender crystals the clamped fre¬ 
quency would be exactly one-half the free-free 
frequency. However for a crystal having finite 
width, only the length is doubled by the mirror; 
the width-length ratio is lessened, and the 
clamped frequency is somewhat higher than 
one-half of the free-free frequency. The limita¬ 
tions on choice of shape are the same as for in¬ 
ertia-drive crystals. 

The suggested upper limits on the width- 
length and thickness-width ratios are approxi¬ 
mate; the chief reason for the limitation is to 
diminish the fluxing of the electric field. The 
dielectric constant in the X and Y directions in 
ADP is about 56, but the piezoelectric constant 
is very small. Thus the dielectric constant en¬ 
courages fluxing but no serious mechanical ef¬ 
fects result so that the effect results only in an 
increase of Qj,. with consequent narrowing of 
band width. 

The dielectric constant of Y-cut RS is 10.0, 
whereas that in the X direction may be hun¬ 
dreds or even thousands depending on field and 
temperature. Thus lines of electric flux may 
bend into the X axis only too easily. In the X 
direction D is nearly as great as in the Y, so 
this fluxing couples in very important unwanted 
properties of X-cut RS. For this reason it is 
more important to keep good shape factors in 
Y-cut RS than it is in Z-cut ADP. 

A second reason for minimizing the shape 
factors is to diminish the losses in glued joints. 
The cemented face of a crystal is subject not 
only to normal forces but also to tangential, 
coupled in both directions by Poisson’s ratio, 
and in the width by piezoelectric coupling. This 
results in a tangential motion of the cemented 
face, of amplitude proportional to the distance 
from center. This tangential motion puts the 
glued joint in shear, a force leading to large 
viscous losses. Obviously these losses are mini- 




234 


DESIGN PROCEDURES 


mized by keeping the face area, particu¬ 
larly the width, as small as possible. 

There is undoubtedly a dependence of the 
electromechanical-coupling coefficient k on 
shape factor, and a maximum may occur for 
some shape. However the effect seems to be 
small, and the above considerations of fluxing 
and viscous loss overshadow it. 

DESIGN OF ARRAY 

Having selected the crystals and the basic 
design, the next step is to lay out an array. This 
really involves two steps: choice of the array 
for directivity pattern purposes and then for 
power handling capacity. 

For the purposes of meeting pattern specifi¬ 
cations one may forget about crystals. The 
transducer is thought of as an array of radiat¬ 
ing surfaces undergoing normal motion. Within 
any one zone the amplitude and phase is usu¬ 
ally considered uniform; various zones are usu¬ 
ally driven at the same phase, but perhaps at 
amplitudes in integral-number ratios of some 
one zonal amplitude. The problem then is to so 
adjust the shapes and relative amplitudes as to 
produce the desired pattern. In doing this one 
makes use of theory which embodies many sim¬ 
plifying assumptions (e.g., for plane arrays: 
an infinite baffle) ; it is very fortunate that 
the majority of transducers do not misbehave 
seriously, and agree rather well with this 
theory. The directivity pattern functions for 
line sources, arcs of circles, squares, circles, 
etc., are available in the literature- and are dis¬ 
cussed in this book (Chapter 4). From these 
an array is selected. 

The second aspect of design of the array is 
the problem of distributing crystals in such 
manner as to produce a radiating surface which 
approximates that selected. Crystals are most 
easily handled if all alike, and it is easiest to 
produce variations of velocity amplitude by 
connecting pairs, triplets, etc., in series, paral¬ 
leling the groups in phase. For this reason the 
array selected should have integral amplitude 
ratios and the number of different zones should 
be a minimum. In fact adequate lobe suppres¬ 
sion can almost always be achieved by just two 
zones of different amplitude. 


Having decided to connect crystals in series 
in certain zones if needed, the crystals are laid 
out in such manner as to facilitate the attach¬ 
ment of foils for electric connection. This lay¬ 
out requires patience, ingenuity, and some prac¬ 
tice to prevent large voltage drops across small 
spaces, etc. In making up the layout the follow¬ 
ing rules are followed. 

1. There should be sufficient crystal area to 
radiate the required power (see Sections 4.4 
and 4.8). 

2. The available area should be only about 60 
to 80 per cent filled by crystals (see Chapter 7). 

3. The crystals should be arranged in groups 
(see Chapter 7). 

4. The center-to-center distance between ad¬ 
jacent groups should not exceed 0.8 wavelength 
(see Section 4.2.4) ; for safety 0.5 wavelength is 
preferable. 

Crystals are rectangular, and are most nat¬ 
urally composed into plane rectangular arrays. 
If a long rectangle is desired, this is very con¬ 
venient. However, if a circularly (axially) sym¬ 
metric pattern is desired this is a temptation to 
use a square array; circular arrays have much 
more desirable patterns than do squares. The 
task of approximating a circle with a number of 
rectangular crystals is unpleasant but justified 
by superior patterns and lobe suppression. 

A typical plane circular array is shown (one 
quadrant) in Figure 10. 

EXTERIOR CASE 

Most commonly the exterior case of a trans¬ 
ducer is steel with a rubber window. If the steel 
is sufficiently thick, corrosion is not a great 
problem. Occasionally bronze cases are used to 
retard corrosion, but if real corrosion resist¬ 
ance is required. Monel metal, passive 18-8 
stainless steel (Type 304 or 316), or passive 
Inconel must be used. Metallic platings such as 
galvanize, cadmium, or nickel actually acceler¬ 
ate corrosion by electrolysis. 

For small volume production cast cases are 
most convenient, but in large quantity forgings 
are likely to be superior. 

Only Monel or passive stainless-steel screws 
should be used, and it is advisable to protect 



EXAMPLES OF VARIOUS STYLES 


235 


them by tar. Socket-head screws are preferable 
to slotted-head. 

Where possible, furnace copper brazing is an 
excellent means of fabricating steel parts; very 
strong watertight joints are easily obtained, 
and a joint may be inspected visually. Such 
bonds must be protected from electrolytic cor¬ 
rosion by tar or by a good paint such as Amer- 
coat (see Section 8.7.5). 

Sound-water [qc] rubber is superior to any 
other for acoustic windows, but a neoprene tire 
stock (40 per cent carbon black) has been used 
successfully in relatively thin flat windows be¬ 
fore plane parallel arrays. 

If gas-filled inertia drive is used it may be 
necessary to stiffen the window. This may be 
done by molding steel bars of appropriate 
shape into the rubber (see Figure 7). Since an 
array should contain crystals grouped together 
with spaces between groups, the steel bars can 
be arranged to occupy these spaces. Usually the 
resulting directivity patterns are not disturbed. 

For symmetric drive units ordinary tin-can 
metal has been used. This use has been limited 
to expendable units in which corrosion is no 
problem, and has been at relatively low fre¬ 
quency (25 kc). Even at this frequency the per¬ 
formance is handicapped a little, and such ma¬ 
terial probably cannot be used much above 50 kc 
without noticeable effect on the transducer. 

In some units a small cylindrical rubber 
sleeve has been used as outer case and window. 
To stiffen the sleeve a perforated metal cylinder 
was forced inside the rubber. This provided 
considerable strength, but at the expense of 
poorer performance. If such a perforated metal 
cylinder is put outside the rubber, care must be 
taken to prevent entrainment of air bubbles; a 
screen of bubbles on the face of a transducer 
virtually prevents radiation of sound. 

The problems of design of the exterior case 
for strength, streamlining, assembly, cost, 
weight, are engineering problems of no great 
complexity and are not rightly part of this book. 

6.7 RESPONSES 

Having selected the kind, size, number, and 
array of crystals, the predicted performance 


(Mason theory) is obtained from the data in 
Section 4.9.8. 

The first step is to evaluate the constant R 
appropriate to the transducer. From this the 
complex impedance without coils is obtained. 
If desired a series tuning coil may be readily 
calculated in. 

Using R, the power radiated by constant volt¬ 
age, with or without coils, and the power radi¬ 
ated when driven by an ideal amplifier are 
easily obtained from the curves in Section 4.9.8. 

If the array is plane the directivity index may 
be estimated from Section 4.4. This will allow 
the calculation of acoustic pressure from the 
radiated power obtained above. Receiver re¬ 
sponses may be obtained from the transmitter 
responses by reciprocity (Section 4.2.3). 

If the predicted response curve does not have 
the desired shape with frequency an equalizing 
network of some kind is required. Preliminary 
estimates of this network may be made, suffi¬ 
cient to allow for design of a container if it is 
necessary. For this the power factor and \Z\ are 
helpful, and may be obtained from Section 4,9.8. 
Final design of such a network (see Chapter 5) 
should await the calibration of the completed 
transducer. 

If the response is flattened by an equalizer it 
is likely that the maximum power output is 
limited by voltage breakdown at the ends of the 
frequency band (see Section 4.8). If this volt¬ 
age is known the maximum possible acoustic 
pressure is easily calculated. However this volt¬ 
age is so dependent upon the care exercised dur¬ 
ing assembly that no estimate can be made here. 

It must be emphasized that the response 
curves predicted in Section 4.9.8 assume the 
transducer to be perfectly efficient. Inefficiency 
will lower the level and often narrows the band 
width, and perhaps introduces extra peaks and 
dips in the curves. 

6 8 EXAMPLES OF VARIOUS STYLES 

It may be helpful to the reader to examine ex¬ 
amples of transducer design. The accompanying 
photographs. Figures 1 through 7, together 
with others in this book, show many of the fea¬ 
tures discussed. No claim is made that these 





236 


DESIGN PROCEDURES 


units are outstanding, but neither are they poor. 
In all instances they are models designed by 
UCDWR. 

Symmetric Drive: CY4, KC2 

Both contain 45° Y-cut RS arranged in a 
single vertical stack; filled with castor oil and 
contained in a tin can. Both operate in the vi¬ 
cinity of 20 kc. See Figure 22. 

Clamped Drive: GA14Z, JB4Z, CPIOZ, 

CQ4Z, CQ8Z, FG8Z 

In every example the transducer contains 45° 
Z-cut ADP crystals cemented to a porcelainized- 
steel backing plate. The motor is contained in a 
heavy steel exterior case with a rubber window. 



Figure 1A. Interior of the KC2 transducer. 


Resonant frequencies range from 24 to 60 kc 
in these examples. It will be noted that the two 
CQ models are dual transducers containing a 
GA14Z and a CPIOZ motor. 

Inertia Drive : FE2Z, GD34Z 

Both units contain 45° Z-cut ADP crystals 
attached to a neoprene window by Cycle-Weld 
cement. The FE2Z was designed to have a broad 
pattern in one direction and a very sharp pat¬ 
tern in the other with excellent lobe suppres¬ 
sion. This was achieved, and the efficiency is 
good. GD34Z was designed as a test of the 
feasibility of strengthening rubber windows by 
steel bars. The results were entirely satisfac¬ 
tory, the efficiency high, and the patterns good. 



Figure IB. Exterior of the KC2 transducer. 




EXAMPLES OF VARIOUS STYLES 


237 


A 


Figure 2A. Interior of the GA14Z transducer. 


B 



Figure 2B. Exterior of the GA14Z transducer. 





Figure 3A. Interior of the CPIOZ transducer. 


Figure 3B. Exterior of the CPIOZ transducer. 

















































238 


DESIGN PROCEDURES 



Figure 4. Interior, exterior, and assembly of the CQ4Z transducer, 





















EXAMPLES OF VARIOUS STYLES 


239 



Figure 5A. Interior of the CQ8Z transducer 


Figure 5B. Exterior of the CQ8Z transducer 



















240 


DESIGN PROCEDURES 



Figure 5C. Assembly of the CQ8Z transducer. 



Figure 7A. Interior of the GD34Z transducer. 




Figure 6A. Assembly of the FG8Z transducer. 


Figure 7B. Exterior of the GD34Z transducer. 




UsK * 


Figure 6B. Assembly of the FG8Z transducer. 



STEEL FRAME RUBBER 


Figure 7C. Cutaway of the GD34Z transducer. 

























































ANALYSIS OF THREE DESIGNS 


241 


ANALYSIS OF THREE DESIGNS 

In this section we review the design pro¬ 
cedures which led to three transducers con¬ 
structed by UCDWR, and we compare the pre¬ 
dicted behavior with observation. It must be 
remembered that one phase of design is the 
process of recalling things which happened long 
ago; this backlog of experience is a prime at¬ 
tribute of any successful designer. It must also 
be borne in mind that these units contain such 
things as foam rubber, the reasons for which 
are discussed in the next chapter. 


Clamped Drive: JB4Z 
Specifications 

This transducer was requested by a UCDWR 
group for use with an experimental echo-rang¬ 
ing sonar system. At first only one unit was 
to be built with a remote possibility of large- 
scale production. To date only one or two more 
have been built. 

It was requested that the array be flat and 
circular roughly 15 in. in diameter so as to re¬ 
semble the JK transducer built by Submarine 
Signal Company. The system was to operate 
over a ±3-kc band width centered at 24 kc, and 
was to radiate very short pings of high inten¬ 
sity. Ping power of 1 kw was contemplated as a 
working hypothesis. No lobe suppression was 
requested and the exterior did not have to be 
streamlined, at least in early models. 

In the first design conference it was evident 
that this would fit naturally into the J-style ex¬ 
terior case designed by UCDWR but resembling 
very closely the spherical JK housing. Some of 
these parts, including the backing plates and 
molded neoprene windows were on hand. 

In this same conference it was agreed that 
series coils tuning at the transducer’s reso¬ 
nance would be advantageous, provided reso¬ 
nance was fairly well-centered in the band. 
Series coils were to be mounted inside the trans¬ 
ducer but would be selected after calibration of 
the transducer to determine its impedance and 
frequency response. 

It was suggested that at a later date horizon¬ 
tal or vertical BDI might be useful, so it was 


decided to wire the four quadrants independ¬ 
ently and make all eight terminals available. 

Choice of Crystals 

The request for very high voltages and acous¬ 
tic intensities clearly indicated the choice of 
ADP. 

Choice of Basic Design 

The J-style case mentioned above was a con¬ 
venient starting point since the parts were on 
hand. This is designed as a backing-plate de¬ 
vice, and, while possible, it would be tedious to 
convert this to inertia drive. In retrospect it ap¬ 
pears that the JB4Z which resulted was quite 
successful; an inertia-driven device would un¬ 
doubtedly have been equally successful. 

Choice of Crystal Size and Shape 

Among the parts on hand were steel backing 
plates % in. thick ready for use. If possible, 
crystals should be chosen to use these plates. 

In Chapter 7 it is stated that resonance will 
occur when a crystal and backing plate satisfy: 

Zc tan 6 = —Zb tan </>. (1) 

In this instance: 

Z, = 5.83 X 10% 

Zb = 39 X 10% 

(j) = 0.381 X i0~®aj. 

There were on hand Z-cut ADP crystals 
lV 2 XV 2 Xbi ill. Ignoring the finite width cor¬ 
rection for the moment, for these crystals 

d = 1.176 X lO-^w. 

Thus the resonant frequency for a 11 / 2 -in. 
long ADP crystal cemented to a %-in. thick 
steel plate would be the smallest value of (o 
which satisfies 

tan (1.176 X lO-'co) = -6.7 tan (0.381 X lO-'co). 
The solution is 

cj = 1.525 X 10% 


/ = 24.3 kc. 

Examination of the finite-width correction 
[see equation (62), Section 3.2.3] indicates that 
for a lV 2 XV 2 -in. crystal on a clamping (%4- 




242 


DESIGN PROCEDURES 


wave) backing plate the finite width lowers the 
resonance roughly V 2 per cent. One might ex¬ 
pect a similar lowering here, since the %-m. 
thick steel plate clamps the crystal fairly well. 
Reducing 24.3 kc by 1/2 per cent, it was pre¬ 
dicted that this system would resonate at ap¬ 
proximately 24.2 kc. 

Design Details 

The parts available allowed an array 14 in. in 
diameter without crowding, and this was ac¬ 
ceptable. The area of the array could thus be 
as great as 750 sq cm, even allowing for wasted 
space. Such an area would easily be capable of 


quadrants and carried through the backing 
plate by backing glands. 

The backing plate was coated with porcelain 
enamel Yiq in. thick which was tested for volt¬ 
age breakdown after grinding. The array was 
assembled in a jig and cemented with bakelite 
cement, being baked under pressure. 

For voltage protection, the silver foils used to 
connect to the crystals were kept in. away 
from the porcelain. Similar precautions were 
observed throughout. For reasons discussed in 
Chapter 7, foam rubber was placed wherever 
required to cover the IVo x i/4-in. crystal 
faces. 



Figure 8. Exterior of the JB4Z transducer. 


1 kw of continuous radiation, since the crystals 
were ADP. It was decided that the crystals 
could be spaced apart. An array was chosen 
which involved 728 crystals, a radiating area 
of 587 sq cm. 

In the chosen array the crystals were ce¬ 
mented together in blocks of three, in phase, 
with intervening foils; each block was thus 
IV 2 X 1/2 in. These blocks were then spaced %2 
in. from each other to fill the 14-in. circle. All 
crystals were in parallel in phase, but leads 
were brought out separately from the four 


The cavity bounded by the backing plate (and 
crystals) and rubber window was filled with 
castor oil. The space behind the backing plate 
was an air cavity which provided the necessary 
low impedance behind the backing plate, and 
also made room for electric connections and 
tuning coils. 

The exterior case itself was cast Meehanite. 
The completed unit formed a 19-in. diameter 
sphere weighing roughly 500 lb, with mounting 
flange to fit a standard Navy column. See Fig¬ 
ures 8 through 10. 







ANALYSIS OF THREE DESIGNS 


243 



§ RUBBER ROD 
PACKING GLANDS 

RUBBER ROD 

CAST MEEHANITE 
CASE 


^ RUBBER ROD 
CRYSTAL ARRAY 
PORCELAIN 
BACKING PLATE 
RUBBER WINDOW 


Figure 9. Cutaway of the interior of a J-style transducer. 


Response 

This unit used essentially clamped ADP crys¬ 
tals so (from Section 4.9.8) : 

R = 7.14 X 10« -4^, (2) 

TlL/xe 

where, Lt = i in., 

= I in., 
n = 728, 

hence, i? = 4.9 X 10\ 

Using this value of R the predicted complex 


impedance was obtained from the graph in Sec¬ 
tion 4.9.8. Since the available calibration data 
report the transducer in series with a series 
coil, such a coil has been calculated into the pre¬ 
diction. This theoretical coil was chosen to 
cancel the reactance at 24.2 kc and was as¬ 
sumed lossless. The predicted impedance 
(dotted) is compared with the observed imped¬ 
ance (solid) in Figure 11. It can be seen that 
the actual transducer resonated a little below 



































































244 


DESIGN PROCEDURES 


24.2 kc, and the coil used cancelled the reactance 
at 23.3 kc. The observed resistance is higher 
than prediction, partly because the coil used did 


The array is of 14-in. diameter so that A = 154 
sq in. At 24 kc the index is —24.9 db. From this 
we conclude that the transducer should produce 



PACKING GLAND 


CRYSTALS ALL IN PARALLEL, 
FOILS RUN FULL LENGTH 


^ X X I ^ CELLTITE ■ FOAM 

RUBBER, AS INDICATED BY 
DARK AREAS 


PACKING GLAND 


PORCELAIN EDGE 


CAST MEEHANITE 
BACKING PLATE 


45* Z CUT ADP CRYSTALS, 
I^X 728 REQ 


Figure 10. One quadrant of the crystal array of the JB4Z transducer. 


not have infinite Q. On the whole the agreement 
of prediction and observation is good. 

From Section 4.4.2 we obtain the expression 
for the directivity index. 


70.8 D — 95.7 db above 1 dyne per sq cm at 1 
m for 1-w input if it is perfectly efficient (see 
Section 4.6.3). Section 4.9.8 tells us the power 
input for 1 v applied with a series tuning coil. 
From this the constant-voltage transmitter re¬ 
sponse can be predicted (in computing this it 






































































ANALYSIS OF THREE DESIGNS 


245 


is necessary to add 20 log a db for the frequency 
dependence of the directivity index). The pre¬ 
dicted curve (dotted) is compared with the ob¬ 
served (solid) in Figure 12. 

Note first that there is a small unexplained 
frequency shift, undoubtedly arising from the 



Figure 11. Complex impedance of the JB4Z 
transducer (solid line) compared with theory 
(dotted line). 


numerous approximations in the theory. Sec¬ 
ond, the efficiency appears to be very low. It is 
not, however, as bad as it appears for these 
reasons. 

1. The typical pattern shown in Figure 14, 
shows that the side lobes are higher than they 
should be. This makes the transducer less direc¬ 
tional. It is estimated that the directivity index 
is approximately —23 db instead of —24.9 at 
24 kc; this lowers the predicted curve 2 db. It is 
true that this is energy going off in an unwanted 


direction, but the cure is in lobe-pattern im¬ 
provement. 

2. Since the actual tuning coil used tuned at 



10 20 30 40 


FREQUENCY IN K C 

Figure 12. Transmitter response of the JB4Z 
transducer with constant voltage applied (solid 
line) compared with theory (dotted line). 



10 20 50 40 50 


FREQUENCY IN KC 

Figure 13. Open-circuit voltage response of the 
JB4Z transducer as a receiver measured at the 
end of a 63-ft cable with a 3-mh coil in series 
with each leg. 

23.3 kc, the frequencies below resonance were 
favored a little at the expense of the response 
at resonance. This is probably sufficient to ex- 




























































246 


DESIGN PROCEDURES 


plain why the two peaks in the observed curve 
are of equal height. 

Applying these corrections to the comparison 
of prediction with observation, it appears that 
the transducer efficiency is —3 db (50 per cent), 
a very reasonable estimate. The maximum open- 
circuit voltage is a couple of db low, confirming 
the estimate of —3 db efficiency. 


0 



Figure 14. Directivity pattern of the JB4Z 
transducer at 24 kc. 


In all respects this is a typical clamped-drive 
unit as constructed by UCDWR; the efficiency is 
only 50 per cent, and the patterns are not quite 
so good as theory. This transducer has been op¬ 
erated for over a year in very high-power serv¬ 
ice with complete satisfaction. 


Inertia Drive: FE2Z 

This transducer was designed and built for 
research by the UCDWR Transducer Labora¬ 
tory, but it has since been used for sonar system 
research. 

The original aim was to test the then new 
technique of Cycle-Welding crystals to rubber 


0 



0 



Figure 15. Directivity patterns of the FE2Z 
transducer at 90 kc. 


windows. As a goal at which to aim, it was de¬ 
cided to attempt to improve the lobe patterns of 







































ANALYSIS OF THREE DESIGNS 


247 


a similar production-type transducer. Figure 15 
indicates the success of this venture. 

Choice of Crystals 

To minimize the size and number of required 
crystals it was decided to build a high-frequency 


rubber would shift the resonant frequency if 
the crystals were too close together. Spacing 
the crystals a little nearly eliminated this. Since 
this shift might introduce some unwanted ef¬ 
fect, it was decided to space these crystals Yiq 
in. apart. Equation (62) of Section 3.2.3 pre- 



Figure 16. Crystal array of the FE2Z transducer. 


transducer; if successful it would be simple to 
design a larger unit to operate at a lower fre¬ 
quency. Some ADP crystals 0.574xl^xl^ in. 



60 70 80 90 100 110 

FREQUENCY IN KC 

Figure 17. Complex impedance of the FE2Z 
transducer (solid line) compared with theory 
(dotted line). 

were in stock and these were used. There was 
no compulsion to resonate at some specified fre¬ 
quency. 

Earlier bench tests had indicated that the 


diets that these crystals should resonate at 95 
kc. However the bench tests indicated a fre¬ 
quency nearer 90 kc when the crystals were 
mounted on the rubber in air, so this was taken 
as the predicted frequency. 

Choice of Array 

The production-type transducer was tall and 
slender, having a pattern shaped like a slice of 
pie, a broad wedge in the horizontal plane, but 
very narrow in the vertical plane. Unfortu¬ 
nately the side lobes were high, and it was this 
drawback which was to be improved in the new 
unit. (See Figure 15.) 

Some old parts for an exterior case were 
available; these would allow an array approxi¬ 
mately 21/2 by 10 in. There was no particular 
specification on the width of the horizontal pat¬ 
tern. An array 21/2 in. wide at 90 kc should pro¬ 
duce a pattern 15 or 20° wide, depending on the 
lobe suppression, so that this was selected. 

At the operating frequency the main lobe was 
to be it 2° wide in the vertical plane at the —3- 
db points. This called for an array 11 in. long, 
but 10 in. was considered to be close enough. 

The array shown in Figure 16 was carefully 
chosen to produce good lobe suppression in the 

























































































248 


DESIGN PROCEDURES 


vertical direction. The ideal velocity distribu¬ 
tion is gaussian, and it was attempted to ap¬ 
proximate this curve in a large number of small 
increments. The lobe suppression is accom¬ 
plished geometrically rather than by putting 


more recent developments it is clear that an 
equally successful unit could be produced using 
thicker rubber. The window used was neoprene 
tire stock; excellent Cycle-Weld cement joints 
to this rubber are easily made, whereas the tech- 



50 60 70 80 90 100 110 120 130 140 

FREQUENCY IN KC 

Figure 18. Transmitter response of the FE2Z transducer with constant voltage applied (solid line) 
compared with theory (dotted line). 


crystals in series. The array is not symmetric 
about the center line, in order to increase the 
number of increments; this should skew the 
patterns negligibly. 

Design Details 

In order to assure uniformity, the crystals 
were tested individually and selected. This step 


niques for oc rubber had not been completely 
worked out at this time. 

The crystals were cemented to the window 
and electrical connections made. Space was pro¬ 
vided for a matching network, and the array 
was bolted into the case. The interior was 
pumped out and filled with Freon gas since this 
has higher dielectric strength than air; Freon 


|70 


^80 


§90 


Figure 19. Open-circuit voltage of the FE2Z transducer as a receiver measured at the end of a 35-ft 
cable. 

is an elaboration which is probably not essential 
to production. 

The exterior case selected was simply a rec¬ 
tangular metal box, with packing glands, to 
which a rubber window could be bolted. The 
window was originally i/o in. thick but was 
ground down to % in. for this unit. In view of 


increases the power-handling capacity but has 
no other effect on the performance. 

Response 

For inertia drive ADP (see Section 4.9.8) 

R = 1.79 X 10« -fy-, 

nLy, 



(3) 


























































ANALYSIS OF THREE DESIGNS 


249 


and in this unit Lt = i in., 
i in., 
n = 100, 

so that R = 8.95 X 10^ 

Using this value of R and assuming the reso¬ 
nance would occur at 90 kc, the predicted im¬ 
pedance was obtained from Section 4.9.8; it is 



Figure 20. Exterior of the FE2Z transducer. 


compared (dotted) with the observed imped¬ 
ance (solid) in Figure 17. The agreement is, on 
the whole, very good; if anything, resonance 
occurred even lower than 90 kc. 

Using the above value of R and the graph in 
Section 4.9.8, the constant-voltage transmitter 



Figure 21. Assembly of the FE2Z transducer. 

response was obtained. This required some rash 
assumptions to obtain the directivity index. 
The expression derived in Section 4.4.2 is usable 
only if the array may be considered an infinite 
plane, a poor approximation for this unit. In 
order to calculate A easily it was assumed that 
an area by %6 in. belonged to each crystal; 
for 100 crystals, A equals 17.6 sq in. At 90 kc 


the wavelength is % in., so that at that fre¬ 
quency the directivity index was assumed to be 
—27.0 db. A correction 10 log a- was applied for 
the frequency dependence of the directivity in¬ 
dex. The response shown (dotted) in Figure 18 
is compared with observation (solid). 

This too indicates that resonance actually oc¬ 
curred a little below 90 kc. Since the theory used 
to obtain the directivity index is inadequate, 
there might be a 1- or 2-db error. Including this 
in our estimate, it appears that the efficiency 
was between —3 and —7 db (not so good as 
gas-filled inertia drive goes.) 

The apparently poor efficiency is explained 
by the construction. In order to prevent exces¬ 
sive deflection of the thin-rubber window from 
hydrostatic pressure, a sheet of foam rubber 
was placed between the back ends of the crys¬ 
tals and the back of the exterior case, in this 
position the crystals were pressed into the Air- 
foam rubber. There may be serious tangential 
loss in the rubber when used this way, and the 
poor efficiency is not surprising. Very probably 
the efficiency would have been 3 db higher had 
the crystals not been standing in foam rubber. 


Symmetric Drive: CY4 
Specifications 

This transducer was to be expendable with a 
service life of roughly hr. Previous to use it 
was to be stored aboard ship in an air-tight 
strong container. It was to be immersed with 
reasonable care and was to operate at approxi¬ 
mately 50-ft depth, although it might occasion¬ 
ally go several hundred feet deep. 

It was to be used only as a projector over as 
wide a frequency band as possible in the vicinity 
of 20 kc. The response was to be as flat as pos¬ 
sible; equalization could be introduced in the 
electric circuit. See Section 5.5. 

The directivity pattern was to be approxi¬ 
mately circular in a horizontal plane, and in the 
vertical plane the main lobe was to be at least 
± 12 ° to 3-db down points at 20 kc (no require¬ 
ments on side lobes). 

The amplifier was to be located nearby and be 
rated at 10 w. Maximum possible acoustic out¬ 
put was requested. Operation was to be sub- 





250 


DESIGN PROCEDURES 


stantially continuous although switching tran¬ 
sients might occur causing momentary high 
voltage. 

Since no cable was needed to reach the ampli¬ 
fier, high impedance was allowed; a few thou¬ 
sand ohms would match the amplifier output 
stage. 

It was essential that the transducer be as 
small as possible. Minimum density was re¬ 
quested. 



Figure 22. Interior and exterior of the CY4 
transducer. 

Mounting facilities were to be worked out as 
design progressed. Immediate large-scale pro¬ 
duction by an inexperienced manufacturer was 
contemplated, and minimum cost was greatly 
desired. 

The entire project was of extreme urgency. 
Choice of Crystal 

Urgency left no choice of crystal. It was evi¬ 
dent that pattern and density requirements 
ruled out backing plates. The ADP crystals 
resonant free-free at this frequency were not 
available in quantity. The required efficiency 
ruled out X-cut RS. The crystals had to be Y-cut 
RS, and preferably in a readily available size. 

Choice of Basic Design 
The nondirectional horizontal pattern invites 
symmetric drive, although it was not immedi¬ 
ately clear that all specifications would be met. 


At this time inertia drive was unproved, and 
symmetric drive was chosen. Similar units 
using X-cut RS had been built previously and 
met the directivity specifications. 

In retrospect it appears that the symmetric 
CY4 is so successful that inertia drive would 
offer no improvement. 

Crystal Size and Shape 

The largest Y-cut RS then stocked in quantity 
by the design group were 11 / 2 x 1 x 1/4 in. These 
crystals resonate at 27.6 kc [equation (63), 
Section 3.2.3] which was too high. However the 
design contemplated had small radiating faces 
(compared with a wavelength) and an appreci¬ 
able radiation inductance was expected. This 
inductance would lower the resonance an un¬ 
known amount, perhaps enough. Longer crys¬ 
tals might be preferable, but they were not 
available. 

Directivity Pattern 

The horizontal pattern of the contemplated 
array cannot be calculated by any available 
theory. However, previous experience indicated 
that a pair of radiating faces 1 in. wide, facing 
in opposite directions in phase, IV 2 in. apart, 
would meet the requirements. Such a pattern 
is quite sensitive to the li/o-in. dimension and it 
was not known if greater separation would do. 
This reinforced the decision to use crystals li/4 
in. long. 

Thus it was decided to use a single vertical 
stack of crystals 11 / 2 x 1 in. of undetermined 
height. Radiation was to be taken from the 
lxi/ 4 -in. crystal faces, and the li/)Xi/ 4 -in. 
faces were to be covered to prevent out-of-phase 
radiation; Corprene or foam neoprene would be 
used for this. For convenience, electric con¬ 
nections would be made underneath this cover¬ 
ing. 

The height required to produce a vertical 
pattern ±12° wide to 3-db down points at 20 
kc is approximately 6 in. This would necessitate 
an outer case 6 I /2 or 7 in. high. It was decided to 
shorten the stack to 5 in. or a little more. A 
line source 5% in. high will produce, at 20 kc, 
a pattern ±15° wide to the 3-db down point.'’ A 

b Compare Figure 45, Chapter 1; the patterns of the 
finished transducers are 15 or 16® wide. 







ANALYSIS OF THREE DESIGNS 


251 


stack of 20 crystals, each in. thick, would be 
the right height so this was tentatively selected, 
final decision depending on the resulting im¬ 
pedance. 

Mechanical Details 

The mechanical design chosen is quite novel. 
It had never been tried before, and there were 
serious doubts about it, but it turned out well 
and has since been used in several other designs 
with equal success. 

The exterior case and acoustic window are 
identical, and consist of a standard olive can. 
This had the advantage that closure is simply 
effected with an ordinary can-sealing device. 
The castor oil is introduced later through a 
soldered bushing, and the hole is closed by a 
pipe plug seated on wet Glyptal.'' Electric leads 
are brought out by a pair of metal-to-glass seals 
soldered into the lid before closure. The entire 
transducer is attached to the rest of the device 
by a crimped seal over a gasket catching the 
rolled edge on the end of the tin can. 

The can material is approximately 0.010 in. 
thick and is reasonably transparent to sound at 
20 kc. There is some evidence that the trans¬ 
ducer would behave a little better in a ^c-rubber 
sleeve. 



10 15 20 25 30 35 


FREQUENCY IN KC 

Figure 23A. Average transmitter response of 
five CY4 transducers (solid line) compared with 
theory (dotted line). 


The tin can would soon rust through and is not 
suitable for prolonged service, but is ideal for 
an expendable device. Corrosion before use is 
prevented by the air-tight packaging. 

When turned over to a manufacturer no great 
difficulty was encountered, and several thou¬ 
sand units were made. 

Response 

From Section 4.9.8 we have 

R = 8.50 X 10« (4) 

For this unit Lt = i in., 

Ly, = 1 in., 
n = 2, 

so that i? = 1.06 X 10^ 

Using this value of R we could obtain a pre¬ 
dicted impedance. However the resonant fre¬ 
quency predicted by Mason theory with the 
width correction is 27.6 kc, whereas the actual 
resonance in the water is lowered to 221/0 kc by 
the radiation reactance. Such a comparison 
would be useless. Instead let us use some hind¬ 
sight, and admit that the frequency is lowered 
this much. If we then compute the predicted 



FREQUENCY IN KC 


Figure 23B. Receiver response of the CY4 
transducer. 


The tin can is soft and easily dented, but m ust 
be smashed considerably before the crystals are 
damaged. There is some compressible material 
inside (Corprene) and the can tends to collapse 
if submerged deeply, but no damage is done. 

c Glyptal insulating varnish supplied by General 
Electric Company. 


impedance from Section 4.9.8 we obtain the 
dotted curves in Figure 23A, B, C, D. We see 
that there is considerable discrepancy which 
would be removed by raising both resistance 
and reactance by a factor of, say, 1.4. 

Examine the consequences of the rather arbi¬ 
trary frequency shift from 27.6 to 221/2 kc. The 















































252 


DESIGN PROCEDURES 


assumption has been made that the crystal re¬ 
mains fully loaded by the radiation impedance 
of water, so that R = 1.06x10®. Of course the 
capacity of Co in the equivalent circuit remains 
unchanged, but the reactance of Co at 221/2 kc 
is greater at the lower frequency. Furthermore, 
in deriving the expressions in Section 4.9 no 


cuit with an added mechanical inductance (as¬ 
sumed constant) sufficient to lower the resonant 
frequency to 221/2 kc. 

The directivity index function derived in Sec¬ 
tion 4.4.2 is not alleged to be usable for an array 
of this kind. However, it is interesting to com¬ 
pare the index it predicts with the index ob- 



transducer (solid line) compared with theory 
(dotted line). 



Figure 23D. Directivity pattern of the CY4 
transducer in the vertical plane. 


width correction was included. As far as Mason 
theory is concerned the resonance should occur 
at 30.9 kc. It is at 30.9 kc that equals —5.55; 
at 221/2 kc should be less by a factor of 1.37. 
Yet we have required it to be —5.55 in comput¬ 
ing this theoretical curve. Thus we see that 
much of the discrepancy results from misuse of 
the theory, and the transducer is not as anoma¬ 
lous as it would appear. 

This situation is not rare, and this example 
serves to illustrate the limitations of the curves 
in Section 4.9.8. A better prediction is obtained 
by calculating directly from the equivalent cir- 


tained by numerical integration of observed 
patterns: 


Frequency 

(kc) 

15 

20 

25 


Observed 

(db) 

— 9.20 

— 9.08 
—10.15 


Predicted 

(db) 

— 8.96 
—11.46 
—13.40 


If we ignore this error, we may proceed to 
calculate a response curve from the theoretical 
index and the data in Section 4.9.8. It happens 
that constant-current transmitter responses are 
available for this transducer so that response 





























































253 


ANALYSIS OF THREE DESIGNS 





















































































254 


DESIGN PROCEDURES 


is computed. See Figure 24A, B, C, D. The 
equation used is: 

db above 1 (dyne/cm^) at 1 m amp = 

70.8 + 10 log ^ ^ j + 10 log + 10 log Rh, (5) 

where 70.8 is the pressure from a 1-w spherical 
source, the directivity index is computed at a 
equals 1 ( 221/2 kc), the a- term is necessary to 
take care of the frequency dependence of P, 
Rki is the power expended when 1 amp flows 
through the unit, and ki is the curve in Section 


4.9.8 appropriate to clamped (symmetric) Y- 
cut RS with R = 1.06 X 

This curve is dotted in Figure 23. The good¬ 
ness of prediction is entirely fortuitous, and 
should not be regarded as a justification of the 
many poor approximations inherent in this 
method. The directivity patterns of a unit such 
as this are fast functions of the dimensions of 
the crystal, and no available theory allows cor¬ 
rection for this in the computations. Further¬ 
more the radiation impedance is complex, and 
changing rapidly with frequency at 20 kc. As 
yet we have no data on the radiation impedance 
(see Section 9.4). 




Chapter 7 

DESIGN ADJUSTMENT 

By T. Finley Burke 


I N THE PRECEDING CHAPTER the steps were out¬ 
lined by which a designer arrives at the 
principal features of a design to meet certain 
specifications. If the Mason circuit w^ere always 
obeyed this would be sufficient. However, there 
remain many features not yet considered which 
influence the behavior, and which must be ac¬ 
counted for if the unit is to be successful. In this 
chapter the aim is to discuss means of minimiz¬ 
ing the deleterious effects so as to finally pro¬ 
duce a unit as good as that predicted in Chapter 
6. Notice that the aim is not to obtain better 
performance, but simply to keep it as good. At 
the present time there is not available sufficient 
information to allow the use of second-order 
effects in such way as to obtain behavior su¬ 
perior to that predicted by Mason theory; it is 
to be hoped that extensive research will, in time, 
discover means of accomplishing this. 

The points to be discussed are understood 
only qualitatively. No adequate theory exists, 
and all that can be done is to point out the ef¬ 
fects, cite examples, and expect the reader to 
acquire sufficient experience to weigh the merits 
of each new problem. Since every decision is a 
compromise of some kind, and since each trans¬ 
ducer is a new mystery, the acquisition of good 
judgment is an expensive and lengthy process; 
the novice finds the conflicting effects confusing, 
but should not be discouraged by poor results. 
Nearly every good transducer in service was 
preceded by a family of failures which, at times, 
did not appear to be a convergent sequence of 
improvements. 

7.1 SINGLE CRYSTALS 

In selecting the size and shape of the indi¬ 
vidual crystals to be used, the first step is to 
adjust the resonant frequency (Section 6.4). 
While sufficient for preliminary designs, this 
may not be the final choice. The remaining prob¬ 
lems are different for the three basic drives. 


Clamped Drive 

The glued surface of a clamped crystal under¬ 
goes tangential motion which results in viscous 
losses in the glue. The amplitude in each direc¬ 
tion is proportional to the crystal dimension. 
Consequently one can expect the efficiency of a 
single crystal to improve as L,^, and are dimin¬ 
ished. Since the efficiency of the whole trans¬ 
ducer is the average of the efficiency of each 
crystal, it is desirable to minimize and 

On the other hand, if and are dimin¬ 
ished the total area of and Ly and L^Ly faces 
must increase since more crystals are required. 
These faces also undergo tangential and normal 
motions, and expend energy in losses. The 
energy so wasted increases with the area of 
these faces, and is thus reduced by increasing 
L,^, and L,. 

The two effects conflict, and some compro¬ 
mise must be reached. Among the transducers 
made by the various suppliers (such as Bell 
Telephone Laboratories, Brush Development 
Company, Submarine Signal Company, and 
University of California Division of War Re¬ 
search [UCDWR]) there is a complete gamut 
of compromise, but the preference seems to be 
Ly > L^o > Lf. Any possible advantages ob¬ 
tained from L,j, > Ly appear to be overshadowed 
by the losses. A convenient shape results and 
good efficiency is obtainable, if the designer at¬ 
tempts to keep L^y ^ However ^ O.IL^ 
seems to be too small; the glued joints to a great 
many small surfaces are too difficult. 

Since the tangential motion in the L^ direc¬ 
tion is less than that in L„., the viscous loss is 
not as severe, and greater L^ could be tolerated. 
However it is important to keep the crystal a 
good parallel-plate condenser, and for this one 
should probably have ^ V 2 ^«- Quite good 
transducers have been built with L, = O.IL,,., 
but this is extreme, and the transducer begins 
to contain too much glue. 


255 



256 


DESIGN ADJUSTMENT 


Inertia Drive 

For gas-filled inertia drive the reasoning is 
quite different. If adjacent crystals are not ce¬ 
mented together (see Section 7.2) negligible 
losses occur on the and L,L,^ faces, so that 
minimum L, and is indicated. However the 
parallel-plate condenser restriction still holds: 
Lf ^ Slender crystals, not cemented to 

each other, are too loosely attached to the win¬ 
dow if and are too small; the crystals can 
lean over irregularly and are too easily dam¬ 
aged. The designer must judge for himself how 
rigid and strong the array must be. 


Symmetric Drive 

Symmetric drive, at least in vertical-stack 
units such as UCDWR Model CY4 (see Figure 
22 of Section 6.9), offers much less freedom of 
choice. In such units a certain directivity pat¬ 
tern is expected at the resonant frequency. The 
pattern is a sensitive function of the crystal 
shape, and allows very small range of selection. 
For example, to produce the CY4 pattern at 20 
kc the crystals must be IV 2 XI in. (so far as is 
now known) ; it is fortunate that such Y-cut 
Rochelle salt [RS] crystals resonate at 221/2 kc. 
To produce an analogous Z-cut ADP stack reso¬ 
nant at 221/^ kc would require crystals perhaps 
21/^xl in.; the patterns would be very different 
and probably not acceptable. Not much has yet 
been done on the dependence of these patterns 
on crystal size and shape except to establish the 
extreme sensitivity. It is certain that no present 
theory allows their calculation, and much more 
experimental investigation is indicated. 

In any event, the selection of shape for ver¬ 
tical-stack design is controlled by pattern re¬ 
quirements, independent of the shape depend¬ 
ence of the losses. There is no particular reason 
to avoid L„, > L^, but L, ^ and ^ V-iL'u 
is indicated. 


Weakened Crystals 

Sometimes it is necessary to produce a reso¬ 
nant frequency lower than can be reached by 


any available crystal. In some instances this 
may be done by weakening the crystal. For ex¬ 
ample, consider 45° Y-cut RS crystals 
11 / 2 x 1 x 14 in. to be used in a vertical stack. In 
air these crystals resonate at 27.6 kc; in a fin¬ 
ished CY4 in water, they resonate at 221/2 kc. 
The radiation is taken from the 1-in. edges. If 
slots are cut into the crystal toward the center 
from the centers of the two li/o-in. edges, the 
free-free resonance is lowered (see Figure 1). 



Figure 1. Crystal weakened to lower its reso¬ 
nant frequency. 


Only shallow slots are required to lower the 
resonance to 22 kc in air, and the resulting 
crystal is quite strong and capable of driving to 
cavitation. At this lowered frequency the radia¬ 
tion reactance is not as effective in lowering the 
resonance in water, so that a finished CY4-style 
unit in water, using these weakened crystals, 
resonates at 18 or 19 kc. This is a rather mild 
example; by extensive slotting and considerable 
weakening the resonant frequency of a 
11/4x1x1/4 in. Y-cut RS crystal was lowered to 
3 kc. 

If the depth of the cut is held constant and 
the slot is widened the frequency varies pecul¬ 
iarly. If the slots were made as wide as L,^ itself 
a slice would have been taken off the sides, re¬ 
ducing the width-length ratio, and the resonant 
frequency would be higher than that of the 
original crystal. Thus one expects the slot to 
lower the resonance, and then as the slot is 
widened the frequency should rise, finally, to a 
higher value. Actually the frequency is constant 
over a considerable range of slot widths, the rise 
coming suddenly just as the slot width becomes 
equal to L^. The reason for this is that the slot 





















GROUPS OF CRYSTALS 


257 


leaves a mass of crystal projecting from each 
of the four corners. Because of “whipping,” 
these masses introduce considerable inertial re¬ 
actance and hold the frequency down. 

If slotted crystals are used in a liquid such as 
castor oil, peculiar effects should arise from the 
oil in the slot. These may be worth investiga¬ 
tion, but it appears likely that the chief effect 
is to restore some of the stiffness removed by 
the slot. To avoid this the slot should be filled 
with an isolation material such as Airfoam 
rubber. 


7 2 GROUPS OF CRYSTALS 

Instead of spacing every crystal away from 
its neighbors or packing all the crystals to¬ 
gether in a block, it is customary to group them 
in small blocks which blocks are then spaced 
apart from their neighbors. The reasons for se¬ 
lecting various block arrangements are differ¬ 
ent for the different drives. 


Clamped Drive 

Two major effects occur when crystals stand 
near or against each other on a backing plate 
and oil or glue is placed between the crystals: 
(1) viscous losses occur because of the lateral 
and tangential motions, and (2) bending forces 
are exerted on the backing plate. 

Considering the first of these, cementing crys¬ 
tals together or coupling them by thin oil layers 
is quite similar to having selected bigger crys¬ 
tals, and the viscous losses at the backing plate 
encouraged by large L,^. and are increased 
(see Section 7.1.1). Furthermore, there appear 
to be losses on the L^^Ly and faces, even 
when the crystals are not yet attached to the 
backing plate. These losses are not at all under¬ 
stood, but probably arise from the fact that no 
crystal face remains perfectly plane as the 
crystal distorts; glued joints between, say, the 
electrode faces (LyLy) are subjected to exten¬ 
sion in some places and compression in others 
resulting in viscous loss. 

Considering the second effect above, the bend¬ 
ing of the backing plate, as a group of crystals 


expands laterally the crystals tend to push each 
other apart. In so doing they bend the backing 
plate convex outward. This is reversed at the 
next half-cycle, and the result is that the flexu¬ 
ral modes of the backing plate are strongly 
driven. In fact even a single crystal expanding 
and contracting tangentially at the plate does 
this too; it is apparent that if a flexural normal 
mode of the backing plate occurs anywhere near 
the driving frequency it is inevitably excited. 

Both of these effects would be minimized by 
spacing each crystal from its neighbor and put¬ 
ting Airfoam rubber between the crystals; the 
effects would be made more troublesome as crys¬ 
tals were crowded together. Unfortunately it is 
not always good to space the crystals apart. For 
one thing, it greatly complicates the assembly of 
a motor having hundreds of crystals. Shims 
must be put in the spaces, and later removed at 
the crystals’ jeopardy. Also, only perhaps half 
of the available area can be covered by crystal 
so that the radiation resistance is unduly low 
[see equation (56), Section 4.2.4]. 

A compromise is effected by grouping into 
blocks containing a small number of crystals. In 
any one block the crystals are glued together 
with the best possible joint. If this joint is poor 
the efficiency is lowered badly. These blocks are 
then lapped flat and cemented to the backing 
plate with intervening spaces. These spaces may 
later be filled with isolation material (see Sec¬ 
tion 7.5.5). 

No fixed rules exist for the choice of crystal 
blocks. It is uncommon to have as many as ten 
crystals in a block; four is perhaps an average 
number. Usually the block is square, and the 
length of the crystals should exceed the other 
dimensions, preferably by as much as a factor 
of 2. If this factor gets too big the crystals are 
long slender fingers, easily knocked off of the 
plate, so that some judgment is needed. 


Inertia Drive 

If gas-filled inertia drive is used there is negli¬ 
gible lateral coupling, even between “touching” 
crystals. Furthermore flexural modes of a rub¬ 
ber window give little or no trouble, so that the 
problems are greatly simplified. If the crystals 



258 


DESIGN ADJUSTMENT 


are spaced apart the radiation resistance is low¬ 
ered and this may be disadvantageous. It is 
probably preferable to bunch the crystals close 
to each other but leave them not cemented. If 
they were cemented the losses occurring at the 
faces would be encouraged to no advan¬ 
tage. 


Symmetric Drive 

To date the only symmetric drive units con¬ 
taining more than a couple of crystals have been 
the UCDWR vertical-stack designs. In them 
there has been no occasion to space the crystals 
in blocks. The procedure has been to pile crys¬ 
tals one on top of the other (on the electrode 
faces) with interposed 0.001-in. foil for elec¬ 
tric connection. The crystals were not cemented 
to each other, and in at least two models the 
foils were wrinkled. The efficiency has always 
been high. 

In an experimental unit flat foils were used 
and the entire pile of crystals was glued to¬ 
gether. The resulting unit had poor efficiency. 
This is not explained, but it is apparently an¬ 
other manifestation of the losses occurring 
when faces are glued together. 


Foiling 

Often blocks of crystals are arranged in 
which the number of crystals per block is equal 
to the ratio of the amplitudes in two lobe-sup¬ 
pression zones. For example, in using a 3 A ve¬ 
locity ratio it is natural to make up blocks of 
triplets. In the low-amplitude zone all three 
crystals are in series in phase and only the ex¬ 
treme foils are driven electrically. One must 
then decide whether or not to place electric con¬ 
ductors at the internal “electrode” faces even 
though they are not connected to anything. 

Two things may be said in favor of having 
these internal electrodes. First, they establish 
equipotential planes and diminish the fluxing 
of the electric field. This is just the matter of 
keeping L, < in order to keep a good con¬ 


denser. Second, these internal electrodes could 
be interconnected among the blocks to help 
maintain uniformity over the array. 

There is an important disadvantage to these 
internal electrodes; the number of individual 
glued joints is doubled. Since losses at these 
joints do occur it might be expected that the 
losses would be increased by the extra joints. 
This has in fact been observed, and it looks as 
if this disadvantage is more serious than the 
advantages above, so that the internal foils are 
not generally recommended. Some thought 
should be given for each new transducer and 
simple tests should be run to determine the Q 
of each style of block in air. 


^ Directivity Patterns 

In selecting the size of a group of crystals to 
form a block the rule given in Section 4.2.4 
must be observed. There it is stated that the 
center-to-center distance between adjacent 
blocks should not exceed 0.8 wavelength. This is 
a fast function, and one does not have to stay 
much inside 0.8 to be safe. However if possible 
it is preferable to try to limit this to 0.5 wave¬ 
length. In a plane array, if this rule is violated 
the system acts like a diffraction grating. The 
directivity patterns resemble those of an optical 
grating, showing first order, second order, etc. 
In transducer nomenclature these higher orders 
constitute side lobes not many db down, and the 
patterns are useless. 


Example 

A typical clamped-drive transducer using 
crystals arranged in blocks is UCDWR Model 
JB4Z in which the crystals are each IV 2 XV 2 X 
in. They are grouped in triplets to form blocks 
1V2X^/^X%. in. In this unit there is no lobe sup¬ 
pression, so that there is a foil between each 
pair of crystals in each block, and all crystals 
are in parallel. The blocks are spaced %2 in. 
from each other and the spaces are filled with 
Airfoam rubber. This unit operates at 24 kc at 



ARRAYS 


259 


which frequency the wavelength is 21/2 in.; the 
greatest center-to-center distance is 1/3 wave¬ 
length. No diffraction-grating behavior is ob¬ 
served near resonance, although it might be ex¬ 
pected near 75 kc. 


unity, and L/\ is likely to be quite small. Thus 
the radiation impedance may be very low for 
such a unit, and even small losses may seriously 
lower the efficiency. 


ARRAYS 

In making up crystal arrangements to pro¬ 
duce desired directivity patterns, several 
sources of trouble may arise. By no means are 
all of these yet discovered but some are dis¬ 
cussed below. 


""" Cylinders 

Sometimes it is required to produce a non- 
directional directivity pattern in one plane and 
to radiate considerable power. A line source, 
such as a vertical stack, would not be capable 
of radiating the power, and a cylindrical source 
is required. This subject is discussed in Section 
4.2.4, from which we note two things. 

A cylinder can be driven by crystals only at a 
finite number of points N around the circle. The 
resulting directivity pattern will have “scal¬ 
lops” corresponding to the N driven points un¬ 
less N > ka. As is noted in Section 4.2.4, the 
patterns become smooth circles very soon as N 
exceeds ka, so that N = ka 2 or 3 yields 
fairly smooth circles. 

When N crystals are arranged in a circle to 
drive a cylindrical source, the total fraction of 
the area driven is AL/2jt(i where L is the width 
of the chord formed by the radiating edge of one 
crystal (usually L,). Then the radiation resist¬ 
ance to be inserted in the equivalent circuit for 
a crystal face is 

j mechanical ohms. 

The factor NL/2na is a particular value of the 
quantity g used in Section 4.4.2. Note the rule 
N > ka above, and that the quantity NL/2m, 
may be written 

(£)(^)- 

For small cylinders N/ka is likely to be close to 


' ^ ^ Lobe Suppression 

Any number of lobe-suppression schemes may 
be invented for a particular shape such as a 
plane circle. The merits of complicated schemes 
must be weighed against ease of construction. 
It is difficult to lay out arrangements for elec¬ 
tric foils if the crystals are not all the same size 
or integral multiples of the same size. Since 
variations of velocity are most easily accom¬ 
plished by connecting varying numbers of crys¬ 
tals in series, lobe-suppression schemes should 
be limited to those having integral-number am¬ 
plitude factors. 

In Section 4.4.1 it is shown that the maximum 
radiated pressure is diminished by lobe suppres¬ 
sion; this diminution resulting from a given 
scheme must be weighed against the lobe-sup¬ 
pression. The 3/1 scheme for circles (diameter 
ratio 0.61) gives —28 db lobe suppression. 
Actually to achieve this requires very nice con¬ 
trol over the phase and amplitude distributions, 
and it is doubtful if much better suppression 
could actually be obtained with any scheme. In 
any event, —28 db is adequate for all ordinary 
purposes, and there appears to be little need of 
schemes involving more than two zones. Al¬ 
though transducers have been built using as 
many as five zones they were not better sup¬ 
pressed than two-zone units. 


^ Multiple Motors 

Very often the desired band width is greater 
than present crystals allow and multiple motors 
are indicated. There are two ways in which the 
motors might be arranged: (1) several separate 
motors near each other, or (2) motors inter¬ 
leaved with each other. Both have drawbacks. 

If separate adjacent motors are used the un¬ 
desired mechanical couplings can be eliminated 



260 


DESIGN ADJUSTMENT 


by suitable baffles. However, the apparent 
source of sound moves around from one motor 
to another as the frequency is changed; in units 
for use as calibration standards this is very 
troublesome. A more important drawback is 
the behavior at the crossover frequencies. If a 
pair of motors for use in frequency bands have 
acceptable patterns when operated alone in 
their respective bands, then if both were to be 
driven at the crossover frequency the pattern of 
combination would be very poor, particularly if 
there were much phase difference between the 
motors. It is unpleasant to disconnect one unit 
and connect the other at crossover, but there 
appears to be no way of avoiding it; the band 
over which crossover troubles occur might be 
made indefinitely narrow by filters, but could 
not be entirely eliminated. 

This crossover difficulty might be overcome 
by interleaving the crystals of the two motors 
so that they occupy the same area. Provided 
phase differences could be fixed up and the 0.8- 
wavelength rule were not violated, this should 
be successful. However this is likely to result in 
serious losses resulting from adjacent crystals 
moving with different velocity. Also much diffi¬ 
culty must be expected from the mechanical 
couplings of the crystals of one motor to those 
of the other. At best the design of such a mul¬ 
tiple unit would be very involved. 

If multiple motors are to be used with cross¬ 
over networks instead of switches, there are 
several ways of connecting them electrically. 
They might be all connected in parallel and then 
tuned with a single coil. However at the fre¬ 
quency at which one motor is resonant the 
others would be shunting capacities, the effec¬ 
tive of the resonant motor would be raised, 
and the band width narrowed. 

A better procedure would be to series-tune 
each motor separately and then parallel the 
units. If the problem is treated by filter theory, 
crossover networks might be designed using the 
inactive elements as filter sections for the active 
motor. This is a very involved problem. 

No matter what crossover network is used, 
the relative phases must be picked correctly to 
give the best patterns at crossover; for this it 
must be remembered that the phase changes by 
Tt crossing each resonance. 


7-4 TANGENTIAL MOTION 

When a surface vibrates tangentially in a 
viscous fluid, shear waves are propagated out¬ 
ward. The attenuation is much greater than 
that of longitudinal waves, and in castor oil at 
frequencies from 10 kc to 100 the l/e distance 
is measured in tenths of millimeters. Crystals 
move tangentially and thus radiate such shear 
waves, and this outgoing energy is a loss which 
reduces the efficiency. 

If a crystal is separated from other surfaces 
by several times the l/e distance it behaves as if 
it were immersed in an infinite ocean of the 
fluid. For this reason crystals only Vs in. apart 
in castor oil are tangentially independent of 
each other. Under this condition the shear 
waves cause a noticeable but not serious loss 
and acceptably high efficiency is achievable. If, 
however, a tangentially vibrating surface is 
placed closer than a shear wavelength to an¬ 
other surface undergoing different motion (or 
stationary) the shear losses are greatly in¬ 
creased, apparently going up like the reciprocal 
of the separation. Thus crystals close to each 
other in castor oil are likely to compose a very 
inefficient array, and crystals not actually ce¬ 
mented together should be spaced apart a milli¬ 
meter or more. If crystals are placed close to 
each other they should be glued, and the glue 
should be thoroughly baked to raise its Q. 

Usually it is difficult to sort out the effects of 
viscous loss in order to demonstrate them, but 
in one instance there is a fairly clear-cut ex¬ 
ample. 

The directivity patterns of a CY4 transducer 
near resonance are acceptable for the original 
purpose, but are by no means circular. In an 
effort to improve the patterns of the CY4, it 
was suggested that more nearly circular pat¬ 
terns would result if every alternate crystal 
were rotated 90° in a CY4 so as to make a four¬ 
faced vertical stack. Of course the lV 2 XVi-in. 
faces of each crystal would have to be covered 
with Corprene to prevent out-of-phase radia¬ 
tion. 

This experimental transducer (XCY8-1) was 
built and tested. The patterns were much im¬ 
proved and were quite usable, being rather 
square but essentially circular instead of re- 



ISOLATION MATERIAL 


261 


sembling figure eight. However the efficiency 
was obviously very poor whereas that of CY4 
is quite good (see Figure 2 in which CY4 is 
shown solid and XCY8-1 dotted). The symp¬ 
toms exhibited by XCY8-1 are typically those 
of a unit of negligible efficiency: open-circuit 
voltage more or less flat in frequency, irregular, 
no marked resonance, and level more than 20 
db below theoretical. 

It was finally realized that the alternate crys¬ 
tals rotated 90° with respect to each other had 
their electrode faces separated only a few thou- 


ISOLATION MATERIAL 
Materials 

In any number of places isolation material is 
needed to control the sound. The ideal sub¬ 
stance, acoustically, is vacuum, but to contain 
the vacuum requires a stiff wall whose charac¬ 
teristic impedance is too high. Thus the next 
best material is a mass of bubbles contained in 
some thin soft-walled substance. To prevent 
soaking up liquids the structure must be a foam 



Figure 2. The receiver response of the CY4 
transducer as compared with that of the XCY8-1 
showing the low efficiency of the latter. 


sandths of an inch, causing serious viscous 
losses. It would, of course, do no good to inter¬ 
pose rigid separators between crystals since the 
losses would then occur between electrode faces 
and separators. What was required was a means 
of separating the crystals by small beads placed 
at the geometric center of each crystal where 
the tangential motion is a minimum. To do this 
disks of solid rubber Ym-m. diameter and % 2 -in. 
thickness were used. To make room for these, 
two crystals were left out, leaving 18. No other 
changes were made and the new unit was desig¬ 
nated XCY8-2. Figure 3 shows the improve¬ 
ment over XCY8-1. There appears to be no 
doubt that the loss in XCY8-1 was caused by 
shear waves in castor oil. 



Figure 3. The receiver response of the XCY8-1 
transducer as compared with that of the XCY8-2 
showing the improved efficiency of the latter. 


rather than a sponge. Even if the input imped¬ 
ance is not zero (vacuum) it must be quite low 
compared with that of water, and should be 
largely reactive to prevent dissipation. In many 
respects this is a description of foam rubber 
whose bubbles contain some gas. Cork is a 
sponge rather than a foam and is prone to slow 
saturation with the liquid unless the surface is 
sealed. To a lesser extent this is also true of Cor- 
prene. 

The input impedance of Cell-tite rubber 
seems to be quite low (not known exactly) and 
at least the normal component is not too resis¬ 
tive. However there appear to be large varia¬ 
tions among samples and some lose the gas by 
diffusion through the walls. This trouble is 














































262 


DESIGN .\DJUSTMEM 


likely to be met in any thin-walled substance, 
but might be less for butyl rubber. 

Queer effects are sometimes observed with 
foam rubber which would be explained by as¬ 
suming a fairly large and perhaps resistive 
tangential impedance. For example, when a 
sheet of Airfoam rubber is cemented to a large 
flat sheet of 20-gauge steel and the reflecthity 
to normally incident plane waves is measured, 
it is found that higher reflectivity is seen look¬ 
ing into the steel with Airfoam on back than 
looking into Airfoam with steel on back. This 
is so generally true that Airfoam is best put on 
a rigid surface through which the sound is in¬ 
cident. 

The Bell Telephone Laboratories have re¬ 
cently used Airfoam rubber jacketed by a fairly 
hea\y layer of rubber. For use in sheets as a 
reflector this should be excellent, but the tan¬ 
gential impedance must be larger. For this rea¬ 
son this jacketed material would not do well be¬ 
tween crystals in an array. 

Corprene is. in many respects, inferior to Air¬ 
foam rubber. There is some reason to think its 
input impedance is higher and its Q lower, thus 
inviting losses in oil. Certainly it exhibits 
marked mechanical hysteresis at very low fre¬ 
quencies. However it is stiffer and more easily 
cut. so that it has its uses in such places as hold¬ 
ing together the vertical-stack transducers. 

Foamglas is a foam whose walls are glass. Its 
density is very low. and it is quite stiff. How¬ 
ever its input impedance is probably* much 
higher than that of foam rubber or Corprene. 
and it does not do as well for isolation. The chief 
use is in places where the isolation must sup¬ 
port its own weight. 


Backing-Plate Terniiiiatioii 

It is usualkv assumevi that a backing plate is 
terminateii by zero impedance. If air is used this 
assumption is justined since the characteristic 
innxHlance of air is onl^v 43 ohms. In many 
transducers the ivgion behind the blocking plate 
is tilled with castor oil and some low-impedance 
lfV.ver is required. Both foam rubber and Cor- 
pix'ue have Ixvn usevi for this, but foam rubber 
is generalb* preferred. In this use the jacketed 


foam rubber developed by Bell Telephone 
ratories should be excellent. 


Case Lining 

Some sound usually leaks out of an oil-filled 
unit in unwanted directions through the metal 
case. To prevent this it is helpful to line toe en¬ 
tire interior with Airfoam rubber or Corprene 
cemented to the case. In this use there is little 
preference of one over the other. 

* Gas-Filled Inertia Drive 

The window of a gas-filled inertia-drive unit 
may be large and thin; if subjected to hydro¬ 
static pressure the lace would bend inwards 
causing various damage. The best way to pre¬ 
vent this is to stiffen the window with steel rods 
molded in the rubber. Occasionally a rigid sur¬ 
face is pro\ided close behind the crystals to pre¬ 
vent their moving backward. Because the crys¬ 
tals pushing against a hard surface might 
break, and an unknown reactive impedance 
might be imposed on them. Airfoam rubber 
is put between the crystals and the rigid sur¬ 
face. In every instance the efficiency is lowered; 
the crystals push rather deeply into the rubber, 
and it seems likely that tangential losses result 
from the crystals rubbing against the Airfoam. 
This practice is not recommended. 


Liquid-Filled Inertia Drive 

In principle inertia drive results if one end of 
the crystal is terminated by a low inqxxiance; 
it is not necessary that the interior Ix' gas-filled. 
Several L CDW R oil-filled transducers have 
been built in which inertia drive was attempted 
by terminating the crystals in AirLwm rublxw. 
In view of Section 7.5.4. the crystals should not 
merely rest on tlie Airfoiim; however, one c.an- 
not Cycle-\^ eld successfully' to .\irt\vun rublx'r. 
Instead the crystals weiv Cycle-Wehhxl to a thin 
strip of solid neopivne and this was in turn 
blocked by the Airfoam. No adoq\iate explana¬ 
tion has been projx\<t\l. but in every instance 






BACKING PLATES 


263 


the efficiency has been very poor. Very probably 
some tangential loss is involved; in any event 
much research would be required to make this 
method successful. 


Spaces between Groups of Crystals 

In Section 7.2.1, reasons were given for 
grouping crystals in blocks which were then 
separated from each other; it was remarked 
that Airfoam rubber may be put in the spaces. 

No rules can be given for when to put Air- 
foam rubber in these spaces. If the transducer 
has acceptable patterns and good efficiency, the 
Airfoam should be left out since it is certain to 
introduce some tangential losses between itself 
and the crystal surfaces. On the other hand, sev¬ 
eral units whose patterns and efficiency were 
bad were cured by the Airfoam. To date the 
procedure is cut-and-try. 

If Airfoam is used it is not necessary to fill 
the gap completely; apparently the mere pro¬ 
vision of a pressure-release surface is sufficient. 
By half filling the gap all the good will be ob¬ 
tained and a little less new loss is introduced. 

For several years UCDWR customarily put 
Corprene between crystals, and even went so 
far as to embed crystals in Corprene on five 
sides. Out of all the units so built only one, GC2, 
was acceptably efficient, and it remains a mys¬ 
tery. There is little doubt that serious tangen¬ 
tial losses occur if Corprene is very near or 
against crystals. 


■ 6 BACKING PLATES 

Throughout the better part of this book a 
backing plate is treated as a simple structure in 
order to facilitate theoretical work. Usually it is 
taken to be a uniform lossless layer backed by 
zero impedance so that its input impedance is 
just -\-jZj. tan cf>. In Section 4.9 it is even re¬ 
garded as infinite at all frequencies. While these 
may be first- or zero-order approximations, a 
much more rigorous treatment is required to 
explain transducers fully. The Naval Research 
Laboratory [NRL] has published a very de¬ 
tailed analysis of certain backing-plate struc- 


tures^ which certainly deserves much study, but 
even there many approximations had to be 
made. 


Spurious Vibrations 

The most common approximation is the as¬ 
sumption of plane waves; any backing plate is 
irregularly excited because the crystals do not 
cover it completely and their surfaces do not 
move as planes. This results in the excitation of 
(probably) every normal mode of the plate, and 
if some normal modes happen to be close to the 
driven frequency they will be strongly excited. 
Except for the limited control over these modes 
described in Section 3.4, nothing can be done 
but cut-and-try if troublesome resonances are 
encountered. As a general rule it is well to cut a 
plate up into pieces small compared with a 
wavelength (in the plate), but this is not always 
possible. Some improvement may be obtained by 
damping the plate, but this cannot lead to high 
efficiency. Of all the backing-plate schemes pro¬ 
posed the NRL unit-construction seems to offer 
the greatest promise (see Section 3.3.4). 


^ Multiple Layers 

The multiple-layer backing plate described 
in Section 3.5 has been tried in a few cases. 
Further investigation of its capabilities seems 
indicated, but it is probably useful only in very 
special problems where space and weight speci¬ 
fications justify its use. 


Glass 

Most backing plates are solid steel, although 
some glass plates have been used (e.g.. Brush 
Development Company Model C-26). Glass has 
much to offer: it is quite strong, has negligible 
dissipation, and is a nonconductor. This last 
property greatly simplifies the problems of re¬ 
ducing stray capacity and handling high volt¬ 
age. No analysis has been carried out, but the 
band width of a glass-backed unit should not be 
much less than that of a steel backed unit. The 



264 


DESIGN ADJUSTMENT 


lower coefficient of thermal expansion should 
allow high-Q glued joints, and any loss of band 
width might be recovered. 


Insulation 

If a metal backing plate is used some insula¬ 
tor should be put between the crystals and the 
metal to reduce the stray capacities and to 
allow high voltages. For this purpose Sub¬ 
marine Signal Company, has used empire cloth. 
Bell Telephone Laboratories has used ceramic 
wafers and mineralized plastic wafers cut in in¬ 
dividual squares to go under each crystal block. 
This is quite good if high-quality glued joints 
are used. UCDWR has used porcelain enamel 
applied in a furnace. This is likely to contain 
pits which must be filled to obtain high voltages, 
the surface must be lapped, and there is some 
question of whether differential thermal expan¬ 
sion might cause exfoliation (to date UCDWR 
has observed none). To offset these disadvan¬ 
tages, one glued joint is eliminated. 


^ Thin Plates 

If a high-impedance material such as steel 
(39X10^) is used for a backing plate it is not 
necessary to use a quarter-wave plate in order 
virtually to clamp the crystal. Plates as thin as 
an eighth wave do quite well. However the reso¬ 
nant frequency is not quite as low as that of a 
completely clamped crystal. If one ignores the 
finite width correction of the crystal (adding 
it later on) the resonant frequency is at least 
close to that which satisfies 

Zc tan d = — Zb tan 0, (1) 

where Zc = characteristic impedance 
of crystal, 

Zb = characteristic impedance 
of backing plate, 

d = in crystal, 

(f> = in backing plate. 

V B 

In deriving this it is assumed that the radia¬ 
tion resistance is zero; it enters slowly for 
the error is not great. 


A graph of this function reveals that co is a 
fast function of when Lg is less than an 
eighth wavelength. This means that for small 
Lg the resonant frequency may vary from crys¬ 
tal to crystal in an array because of small de¬ 
partures from uniformity. For this reason very 
thin plates should not be used. 

An eighth-wave plate reduces the frequency 
almost to that of a quarter-wave plate for 
Zg » Zp. For plates thicker than quarter wave 
the resonance moves down very slowly and the 
Q of the resonance rises rapidly. Such very 
thick plates are not recommended. Fry, Taylor, 
and Hen vis discuss this subject more thor¬ 
oughly.^ 


-- WINDOWS 

The ideal acoustic window for transducers 
does not exist; it would be identical with sea 
water in all respects except to be an electric in¬ 
sulator and a solid. Unfortunately all real win¬ 
dows differ significantly from water. For one 
thing the slope of the characteristic impedance 
of water versus temperature is anomalous and 
has opposite sign from most substances. Thus 
any window can be expected to match water 
only at one temperature. Fortunately it is not 
necessary to match perfectly for most uses, and 
in fact a decided mismatch is often tolerable. 


Sound-W ater (Qc) Riil)ber 

For several years a rubber developed at NRL 
and available from B. F. Goodrich Company 
has been used for windows. It is commonly 
called oc rubber, and until recently that title 
was a unique description. Recently Bell Tele¬ 
phone Laboratories [BTL] has developed their 
own QC rubber and the term is often used indis¬ 
criminately. Acoustically there is little differ¬ 
ence between the two. The BTL rubber appears 
to be a little stronger and harder, but no im¬ 
portant differences are claimed. 

In both rubbers it has been attempted to 
match the o and c of water separately as well as 
the QC product. The success is remarkable, and 
there is no doubt that these rubbers are the best 




CROSSTALK 


265 


window material available; quite thick curved 
sections may be used with negligible effect on 
patterns or response. 

Both rubbers bond to metal and in this way 
make strong (but flexible) windows. 


Neoprene 

During World War II UCDWR used neoprene 
for windows to facilitate procurement; tests 
were run which indicated negligible harmful ef¬ 
fect in ordinary circumstances, and the neo¬ 
prene windows were adopted to the exclusion of 
QC rubber. However difficulty was encountered 
in a unit having a cylindrical rubber window 
IOV 2 in. inside by 1-in. wall thickness at 42 kc. 
When QC rubber was substituted the troubles 
disappeared. For thin (say V 2 -in.) windows in 
flat sheets before plane arrays, neoprene is 
acoustically satisfactory and mechanically su¬ 
perior to QC rubber, provided effects of the order 
of 1 db are ignored, but in thick sections and 
particularly in curved thick sections neoprene 
is unsatisfactory. 


Other Materials 

Many other materials are usable for windows 
in special applications. Some of these are dis¬ 
cussed in Section 3.7 and in Chapter 8. 


Size of Window 

Ideally a window should probably extend all 
the way around a transducer in order not to dis¬ 
turb the directivity patterns. Bell Telephone 
Laboratories particularly tend to extend their 
QC-rubber windows about ±90° from the for¬ 
ward direction. However most other manufac¬ 
turers usually use a window which is just a 
little bigger than the crystal motor and not too 
far away. Probably some shadowing results but 
it does not appear to alter the directivity pat¬ 
terns appreciably. By restricting the size of the 


window greater strength is obtained; an im¬ 
portant consideration in many units. 


"« CROSSTALK 

In some very special applications it is neces¬ 
sary to operate a transmitter and a receiver 
near each other, simultaneously, at the same 
frequency. Such a system is troubled by cross¬ 
talk, the signal which passes from projector to 
receiver through close-in or internal paths. 

To reduce crosstalk it is necessary, of course, 
to keep the receiver out of the main lobes of the 
transmitter directivity pattern. However this is 
usually not enough since the two motors are 
mounted in a common structure. It has been 
found that sound will run through a metallic 
structure just like an electric current. If a 
metallic path exists by which sound may get 
from one motor to the other it will do so, no 
matter how unbelievably devious the path. It is 
usual that the most important crosstalk paths 
run through washers to bolts, from bolts to 
angle iron, and from the angle iron to a casting, 
and so on, perhaps through a dozen compart¬ 
ments. 

It is necessary to open these paths by insert¬ 
ing isolation material such as Corprene. When 
isolating a bolt it is necessary to provide not 
only Corprene washers under the head and un¬ 
der the nut, but also a Corprene bushing be¬ 
tween the bolt and the drilled hole. Foam rubber 
is not recommended for this use because it takes 
a permanent set too easily under static load. 
Corprene is stiffer but also will take a set, so 
that very large areas should be used to distri¬ 
bute the load. 

In tightening isolated bolts, lock nuts should 
be used so that the nut can be left loose; too 
much compression of Corprene restores a sig¬ 
nificant amount of crosstalk. Isolation blocks 
should be provided immediately adjacent to 
both the receiver and the transmitter. It is 
helpful also to support the crystal motors within 
the cases on isolation material. 

If the two motors are mounted in a common 
exterior case it is necessary to so design the 
interior that any oil paths involve devious 
routes. It is helpful to connect the two chambers 




266 


DESIGN ADJUSTMENT 


only by tubes or holes whose diameters are 
small compared to a wavelength. 

If, in the common case, the two motors share 
a large acoustic window it is found that the win¬ 
dow material can act as a wave guide, piping 
significant amounts of energy from one unit to 
the other. This happens even for qc rubber win¬ 
dows because, while close, this material is not 
identical with water. If this coupling exists it is 
necessary to insert a large impedance mis¬ 
match in the window material between the two 
motors. In rubber this is best accomplished by 
molding in a Corprene layer, or by molding in a 
quarter-wave steel layer. 

If two motors occupy a common exterior case 


it may be important to use a oc-rubber window 
on the transmitter. While the transmission 
through other windows might be as high as 99 
per cent, a little of the transmitted energy fails 
to escape and returns to the interior. This rep¬ 
resents a negligible loss as far as transmission 
is concerned, but the returned energy may in¬ 
crease the internal energy density greatly, thus 
raising the crosstalk level by internal paths a 
great deal. 

If every precaution is taken to reduce cross¬ 
talk, it is usually possible to reduce the level be¬ 
low that determined by long-range scattering 
(reverberation). This is, of course, a lower 
limit beyond which there is no point in striving. 



Chapter 8 

CONSTRUCTION TECHNIQUES AND EQUIPMENT 

By Fred M. Uber 


INTRODUCTION 

T he numerous experimental and produc¬ 
tion techniques involved in the construction 
of transducers have resulted from the efforts of 
many different individuals working in various 
laboratories. In order to present the best meth¬ 
ods there has been no attempt to limit the sub¬ 
ject material to the practices of any one indi¬ 
vidual laboratory. In particular, it has been felt 
that a limitation of this chapter to a discussion 
of the techniques in use at University of Cali¬ 
fornia Division of War Research [UCDWR] 
would not be considered in the best interest of 
the transducer art. Consequently, the writer has 
made a special effort to visit the several labora¬ 
tories engaged in research and development on 
synthetic crystal transducers in order to be able 
to evaluate critically or at least describe the 
various techniques in current use. 

Although it is obviously impossible in a chap¬ 
ter of this kind always to give proper credit 
to the originators of various techniques and 
items of equipment, an effort will be made to 
do so, at least in so far as credit is due the 
various laboratories. Where photographs have 
been reproduced, by-lines indicating their 
source will occur in the titles to the figures. 
Besides UCDWR, information has been ob¬ 
tained principally from the Brush Development 
Company, the Bell Telephone Laboratories 
[BTL], and the Naval Research Laboratory 
[NRL]. 

Precautions When Handling Crystals 

The need for exercising care in the handling 
of piezoelectric crystals should be apparent to 
those familiar with their function. If not, the 
discussion on properties of crystals, which fol¬ 
lows in the next several sections of this chapter, 
should make it clear. The electrical character¬ 
istics of crystals in particular can be adversely 
affected if the crystals are improperly handled. 
Of all the variables and unknown quantities 


that are encountered in the construction of syn¬ 
thetic crystal transducers perhaps the most 
annoying one is associated with the irresistible 
urge of individuals to handle the crystals with 
their bare hands. To one accustomed to dealing 
with inanimate objects this may not appear off¬ 
hand as an insurmountable difficulty, but in 
actual practice in the laboratory or in small- 
scale production it becomes almost impossible 
to control. From the time work is begun on an 
original mother crystal until a transducer is 
completed, numerous operations must be per¬ 
formed, including final surfacing operations, 
application of electrodes, polarizing, cementing 
to supporting structures, wiring, and final 
testing. This entire process may require for its 
completion periods of time varying from a few 
days to a few weeks and perhaps as many as 
a dozen individuals may have taken part. When 
the transducer has been finally assembled it is 
difficult, if not impossible, to be certain that one 
or more of perhaps a few hundred crystals have 
not been touched by someone’s fingers. 

The actual chemical and/or physical changes 
that are brought about on the surface of a 
crystal due to human contact can readily be 
imagined although they may defy detailed sci¬ 
entific description. Both Rochelle salt [RS] and 
ammonium dihydrogen phosphate [ADP] are 
water soluble so that any contact with moisture 
would certainly produce a deleterious effect on 
the surface conductivity. It is quite possible 
that materials which may be deposited on the 
crystal surface are more hygroscopic in nature 
than either ADP or RS. Substances present on 
the skin may also react chemically with the 
crystalline materials. There is apparently no 
point in trying to pursue these suppositions in 
detail, particularly since there seems to be but 
one solution to the entire problem. This solution 
is simply to refrain from touching crystals with 
the bare hands. 

Various laboratories have furnished protec¬ 
tive coverings, either for particular fingers or 


267 


268 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


for all of the fingers on both hands. At the 
Brush Development Company, where numerous 
production workers are engaged in the manu¬ 
facture of bimorph crystals, a relatively high 
degree of success has been attained by requiring 
that rubber finger cots be worn. In their ex¬ 
perience it was not sufficient to have these cots 
worn on the thumb and index finger, which 
alone are normally used in handling crystals. 
When this was done, some individuals would 
resort to using the unprotected fingers. Hence, 
their current practice is to require that these 
cots be worn on all fingers of both hands. Even 
this does not constitute a sufficient solution un¬ 
less the rubber cots are constantly maintained 
in an immaculate condition. Just how difficult 
this can be will be realized when one recalls 
how frequently the fingers may be allowed to 
come in contact with other parts of the body. 
Objections to the use of rubber coverings on 
the fingers arise as a result of the enhanced 
perspiration beneath them. An attempt to over¬ 
come this objection has led to a trial of fairly 
thick rubber finger protectors containing per¬ 
forations but they are not a satisfactory solu¬ 
tion. Cotton gloves, such as are used by film 
cutters in the motion-picture industry, have 
also been tried. 

In the last analysis it would appear that the 
eternal vigilance requisite for the successful 
production of high-power crystal transducers is 
probably to be expected only of that relatively 
small number of individuals who are fully cog¬ 
nizant of the factors involved and who are 
capable of painstaking work. However, there 
are many auxiliary aids which may prove effi¬ 
cacious in stimulating satisfactory performance 
of tedious work. For one thing, great impor¬ 
tance must be assigned to psychological factors. 
One scheme, so advantageously exploited among 
nurses and workers generally having to do with 
public health, is the suggestive use of white 
to promote cleanliness. Another psychological 
stimulus could be furnished by providing each 
operator with an air-conditioned booth or small 
room in which to work. 

The effect of adverse humidity conditions on 
both RS and ADP crystals makes it appear ad¬ 
visable to have a special air-conditioned room 
in which the crystal processing and the assem¬ 


bly of crystals into transducers can take place. 
Equipment should be provided which filters the 
air in addition to controlling the humidity and 
temperature conditions. Related to this problem 
of air conditioning is the further problem of 
providing for the disposal of the dust which 
arises during dry-grinding and milling opera¬ 
tions. This is discussed further in Section 8.4.2. 
The use of adhesives which contain volatile or¬ 
ganic solvents may also make provision for an 
adequate chemical hood advisable for spraying 
operations. 


« 2 CHARACTERISTICS OF CRYSTALS 
^ Chemical Properties of RS 

Chemically, RS is a double tartrate of 
potassium and sodium having the formula: 
KNaC4H406*4H20. Rochelle salt crystals have 



Figure 1. Stability limits of Rochelle salt as a 
function of temperature and relative humidity. 


been known for nearly three centuries, having 
been produced from the dextro form of tartaric 
acid which was abundantly available commer¬ 
cially from the wine industry. The fact that RS 
contains four molecules of water of crystalliza¬ 
tion has an important bearing upon its general 
behavior. At 55.6 C two molecules of RS trans¬ 
form into one molecule of sodium tartrate and 
one of potassium tartrate with the evolution 
of one molecule of water. Complete liquefaction 


























CHARACTERISTICS OF CRYSTALS 


269 


results when the temperature reaches 58 C. The 
behavior at temperatures below 50 C can be 
seen by an inspection of the curves in Figure 1. 
At relative humidities higher than the upper 
limit shown in this curve, moisture will collect 
on the surface of the crystal, while at relative 
humidities below the lower limit given in the 
curve, the crystals dehydrate. The maintenance 
of proper storage conditions for RS crystals is 
discussed at some length in Section 8.2.9. 

The solubility of RS in water at 0 C is 420 g 
per 1 and at 30 C it is 1,390 g per 1. This high 
solubility is an important consideration in con¬ 
trolling the rate of crystallization of RS from 
saturated solutions by decreasing the temper¬ 
ature. Rochelle salt is only slightly soluble in 
ethyl alcohol so that absolute alcohol may be 
used to remove surface moisture from crystals. 
However, dehydration would result if absolute 
alcohol were used too generously or remained 
in contact over too long a period. Rochelle salt 
is soluble in ethylene glycol, but not in benzene, 
carbon tetrachloride, and in numerous other 
organic solvents. 

The density of RS at 25 C is 1.775 ± 0.003 g 
per cu cm. 


Crystallizing RS Bars 

Considerable information exists in technical 
literature on the growth of crystals from aque¬ 
ous solutions, both on a laboratory and on a 
production scale. Successful crystallization of 
large bars depends on several factors. 

1. There must be a carefully controlled de¬ 
gree of supersaturation of the solution in order 
to provide the highest rate of deposition of 
crystalline material without the formation of 
flaws or irregular growth. This control of the 
supersaturation may be brought about in any 
one of three ways. The first one is essentially 
of laboratory importance only and consists of 
controlling the rate of evaporation while main¬ 
taining a constant temperature. The rate of 
flow of water vapor from the solution into a 
condensation trap can be regulated by varying 
the air pressure in the system. The second 
method^- ^ controls the saturation by a variation 
in the temperature and has been employed on 


a large scale in the commercial production of 
RS. With this method, it is customary to start 
with a high initial temperature and allow the 
solution to cool at a carefully controlled rate 
for a period of 3 to 6 weeks. The initial tem¬ 
perature must not be above 40 C or sodium tar¬ 
trate will be deposited. It is convenient if the 
temperature finally reached is approximately 
that of a normal working room so that the 
crystals do not crack upon being brought out 
of the growing room. The third method^ of 
controlling the concentration depends on circu¬ 
lation of the solution between a stock of finely 
divided crystal material and the growing bars. 
In this case a temperature differential is main¬ 
tained to promote the dissolution of the finely 
divided crystals and the subsequent deposition 
on the growing bars. 

2 . Seeds of proper shape and orientation 
must be provided. In the case of RS, these seed 
bars may have a square cross section % to V 2 in. 
on a side and may be as much as 20 in. long. 
They are placed in slots on the bottom of the 
rocking tank which contains the solution. The 
orientation of the seed bar in the slot depends 
on whether X-cut or Y-cut crystals are to be 
processed from the final mother bar. In Fig¬ 
ure 7 the original seed bar is shown being milled 
off the mother crystal in order to provide a 
reference surface. 

3. It is important to provide for a continual 
circulation of the solution over the entire sur¬ 
face of the growing bar. Both the rapidity of 
the circulation and its direction of motion are 
fairly critical. In the absence of satisfactory 
circulation conditions, the rate of crystal growth 
is not uniform over the surface of the bar so 
that vacant or flawed regions known as veils 
occur. Other factors, such as the hydrogen-ion 
concentration of the solution, also have an im¬ 
portant bearing on the rate and quality of 
crystal growth. 

Synthetic RS crystals have been produced in 
this country in commercial quantities for many 
years by the facilities of the Brush Development 
Company and the crystals employed at UCDWR 
during World War II were made available by 
them. The writer of this chapter has had no 
direct experience in the growing of RS crystals, 
but has had the privilege of visiting the plant 



270 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


of the Brush Development Company and ob¬ 
serving the growing and processing techniques. 


8.2.3 Thermal Behavior of RS 

The thermal-expansion coefficients of RS 
have been reported by Vigness^ to have the 
following values parallel to the direction of the 
axes. From the standpoint of construction tech¬ 
nique, the thermal-expansion coefficients are of 
interest only as they affect the differential ex¬ 
pansion between a crystal and the base to which 
it is cemented. 


Axis 

Temperature 

range 

in degrees C 

Coefficient 
per degree C 

X 

12-35 

58.3 X 10-c 

Y 

12-24 

35.5 X 10-6 


24-35 

39.7 X 10-6 

Z 

14-24 

42.1 X 10-6 


24-35 

34.6 X 10-6 


The variation with temperature of some other 
physical properties of RS will be found dis¬ 
cussed elsewhere in this volume. Its electrical 
resistance as a function of temperature is 
treated in Section 8.2.4. 


Electrical Properties of RS 

The characteristic curve of leakage resistance 
as a function of temperature for an RS crystal 
is shown in Figure 2. The measurements for 
this graph were obtained by McSkimin“ and 
were taken on a crystal 1.6 cm long, 1.0 cm wide, 
and 0.4 cm thick with tin foil electrodes being 
placed on the largest surface. Hence, the re¬ 
sultant interelectrode distance was 0.4 cm. It 
will be noted that the leakage resistance de¬ 
creases very rapidly with increase in temper¬ 
ature above 43 C. In obtaining these data the 
relative humidity was maintained at the equilib¬ 
rium point for the dehydrated salt, that is, at 
that value of the relative humidity where dehy¬ 
dration just begins. The entire crystal was 
immersed in oil. At a temperature of 51 C the 
leakage resistance has decreased to 500,000 
ohms. A further increase in temperature results 


in still further leakage, until at a temperature 
above 55 C the crystal is rendered useless be¬ 
cause of melting. 

If the temperature of an RS crystal is either 
lowered or raised rapidly, leakage resistance 
curves differing widely from the characteristic 
curve of Figure 2 may be obtained. As an ex¬ 
ample of this behavior, a crystal coated lightly 
with Vulcalock cement was placed inside a can, 
together with a small quantity of crushed hy¬ 
drated RS. It was found by McSkimin^ that 
raising the temperature of such a crystal only 
1 degree caused the leakage resistance to drop 
very rapidly from over 100 megohms to several 
megohms as shown in Figure 3. At 35 C the 
crystal had less than 100,000 ohms resistance. 
The only satisfactory explanation of this be¬ 
havior appears to be that RS adsorbs water on 
its surfaces. The amount of water adsorbed is 
a function of both pressure and temperature: 
the higher the temperature the less the adsorp¬ 
tion, the higher the pressure the greater the 
adsorption. On this theory the behavior of the 
Vulcalock-coated crystal can be explained by 
assuming that a small quantity of adsorbed 
water was trapped beneath the cement on the 
crystal surface. Raising the temperature only 
slightly could result in driving off a small 
amount of this water into the very limited 
space between the cement coating and the crys¬ 
tal surface with a resultant rapid increase in 
the relative humidity. The high relative humid¬ 
ity resulted in the excessive leakage measured. 

By removing any adsorbed water from the 
crystal before putting on the surface coating, 
it should be possible to approach quite closely 
the characteristic curve of Figure 2. This pro¬ 
cedure was followed to obtain the second curve 
shown at the right in Figure 3. The tremendous 
improvement is very marked. In numerous tests 
it was found by McSkimin® that the removal of 
adsorbed water minimized leakage in all cases. 

For transducer applications where it is nec¬ 
essary to apply high voltages to the crystal 
electrodes, the presence of moisture on the 
interelectrode surfaces of RS crystals will cause 
arcing. Arcing between electrodes will occur at 
some critical voltage, the failure taking place 
quite rapidly. This voltage breakdown may be 
caused to occur at much higher potentials by 








CHARACTERISTICS OF CRYSTALS 


271 


using care in the removal of adsorbed water. 
McSkimin^ has reported that a crystal having 
several hundred megohms leakage resistance 
may nevertheless have a small amount of ad¬ 
sorbed water on its interelectrode surface. If 
the crystal is immersed in oil or if the surface 
is cemented over and perhaps attached to other 
materials, this trapped water cannot be dis¬ 
persed readily in the case of a sudden increase 



Figure 2. Characteristic leakage resistance of 
Rochelle salt crystals. 


in temperature. Owing to the power dissipation 
by even a high leakage resistance, the temper¬ 
ature at the surface of the crystal will be in¬ 
creased slightly. As has already been pointed 
out in discussing the curves of Figure 3, a tem¬ 
perature increase of only a degree or so may 
be necessary to cause a rapid decrease in re¬ 
sistance. Since the power dissipation increases 
rapidly with a decrease in resistance, the effect 
is seen to be regenerative and voltage break¬ 
down happens abruptly. The importance of re¬ 
moving all adsorbed water is evident. 

Another application in which it is of extreme 


importance to keep electrical leakage at a mini¬ 
mum is where it is desired to use RS in trans¬ 
ducers operating at frequencies of only a few 
cycles per second. The performance of a trans¬ 
ducer at low frequencies is limited seriously by 
a low-leakage resistance so that unusual pre¬ 
cautions must be taken in such equipment to 
obtain the highest possible values of electrical 
resistance. 



Figure 3. Leakage resistance of a Rochelle salt 
crystal coated with Vulcalock cement, with and 
without adsorbed water. 


® Chemical Properties of ADP 

Piezoelectric ADP crystals are composed of 
ammonium dihydrogen phosphate, NH4H2PO4, 
also called primary ammonium phosphate. In 
literature'^ of the Brush Development Company, 
these crystals are referred to as PN crystals. 
ADP crystals are thus seen to be without any 
water of crystallization and are therefore sta¬ 
ble in vacuum, in air up to 93 per cent relative 
humidity, and in the presence of strong desic¬ 
cants. The density of ADP is 1.803 g per cu cm. 

ADP crystals are readily soluble in water. 






























































































272 


COrsSTRUCTION TECHNIQUES AND EQUIPMENT 


The number of grams of ADP per 1,000 g of 
saturated solution at any temperature T in 
degrees centigrade, can be calculated from the 
following expression; 171 + 4.70 • T. A large 
number of organic solvents are without notice¬ 
able effect on ADP, including carbon tetra¬ 
chloride, the lower ketones and esters, and ben¬ 
zene. It is not soluble in castor oil. 

At temperatures higher than 125 C, an ADP 
crystal will lose ammonia from its surface un¬ 
less kept in an atmosphere containing ammonia 
vapor. At 125 C the dissociation pressure is 
0.05 mm Hg; at 150 C it is still below 1 mm of 
Hg pressure. Continued loss of ammonia results 
eventually in a coating of phosphoric acid on 
the crystal surfaces. 


Crystallizing ADP Bars 

The production of synthetic ADP bars has 
been carried out on either a pilot plant or com¬ 
mercial scale during World War II by four 
different laboratories. Pilot plants were in op- 



Figure 4. Large ADP crystal grow in recipro¬ 
cating radial crystallizer. (Bell Telephone Labo¬ 
ratories.) 


eration at NRL and at BTL. Commercial grow¬ 
ing plants, financed by the government, were 
built and operated by the Brush Development 
Company at Cleveland and by the Western Elec¬ 
tric Company at its Hawthorne Plant. No ADP 
bars have been crystallized at UCDWR and no 
direct experience with growing processes, other 
than brief observation, has been had by the 
writer. Consequently, the intention here is not 
to enter into a detailed discussion on the growth 


of ADP bars but rather to outline the general 
steps in the process and to indicate where fur¬ 
ther information may be found. The most de¬ 
tailed report available on the crj’stallization of 
ADP bars is contained in a confidential publi¬ 
cation of BTL." This report covers the labora¬ 
tory work which was done at BTL in connection 
with the design and operation of the growing 
plant operated by the Western Electric Com¬ 
pany. It will be assumed here that anyone inter¬ 
ested in growing ADP bars will become familiar 
with the contents of this report. 

The flat seed plates for growing ADP crystals 
are square and have about the same area of 
cross section as the fully grown mother bar. 
The seed plate can be seen in the central section 
of the mother crystal photographed in Figure 4. 
A seed plate, fully capped, is also shown sche¬ 
matically in the drawing of Figure 9. These 
seed plates are obtained by slicing the full 
grown bars. Since the gi’owth of ADP crystals 
is primarily in the longitudinal direction, it is 
difficult initially to obtain seed plates of large 
area. In fact, the greatest deterrent to the rapid 
inauguration of a growing plant is the time- 
consuming task of producing the initial supply 
of seed plates with large cross sections. 

Two types of equipment have been used for 
growing ADP bars. One contains horizontal 
trays which are subjected to a rocking motion. 
The rocking-type crystallizer had previously 
rendered excellent service in the synthesis of 
RS bars; most of the ADP bars produced to 
date have come also from the rocking type of 
equipment. The second type of apparatus, so far 
confined largely to pilot-plant operations, pro¬ 
vides circulation by having the seed bars un¬ 
dergo a reciprocating rotary motion inside a 
cylindrical solution tank. The latter t\*pe plant 
seems to possess some superior advantages for 
growing ADP bars. 

The supersaturation of the ADP solution may 
be maintained either by gradually decreasing 
the temperature of the solution or by the con¬ 
tinual addition of salt to a constant temperature 
bath. The initial temperature of the saturated 
solution at the Hawthorne Plant was 41 C for 
the seed-capping operation and 46 C for the 
subsequent growth of the seed caps to mature 
bars. The final solution temperature should be 





CHARACTERISTICS OF CRYSTALS 


273 


that of the processing room in order to avoid 
fracture of the crystals by introducing thermal 
strains upon their removal from the growing 
tanks. Other things being equal, the crystals 
should preferably be grown by means of a 
constant-temperature process, but operating 
controls seem to be somewhat more difficult. If 
technical difficulties could be satisfactorily over¬ 
come, the ideal method would be a continuous 
constant-temperature operation in which the 
crystal bar would be withdrawn from the satu¬ 
rated solution at exactly the same rate as 
formed. 

The production rate and the quality of the 
ADP bars depends markedly upon the purity of 
the solution and difficulty has been experienced 
in getting sufficiently pure materials. The im¬ 
purity that causes the greatest concern is the 
sulphate ion and one must go to great lengths 
to reduce its concentration to an optimum value. 
Its bad effect on the electrical resistivity of the 
bars will be discussed in Sections 8 . 2.8 and 8.5.7. 
On the other hand, the rate of growth of ADP 
bars is very materially improved by the presence 
of the sulphate ion. Particularly in the capping 
of flat seed plates, advantage has been taken of 
this fact to hasten the capping process. After 
complete caps have been formed, the seed crys¬ 
tals should be transferred to a solution rela¬ 
tively free from sulphate, the exact sulphate 
concentration depending on the electrical re¬ 
sistivity desired in the final product. Of the 
metallic impurities, barium ions may give some 
trouble. For details, refer to the original re¬ 
port.'^ 

8 .2.7 Thermal Behavior of ADP 

The melting point of ADP crystals is 190 C. 
There is no transformation point. Curie point, 
or other thermal irregularity between the melt¬ 
ing point and —100 C. Decomposition with loss 
of ammonia may occur below the melting point 
and is discussed in Section 8.2.5. 

The thermal coefficients of expansion for 
ADP have been determined and are (33 ±3) X 
10^® per degree C, perpendicular to the Z axis, 
and (5 ± 3 ) X 10“^^ per degree C, parallel to the 
Z (optic-) axis. Sudden cooling of ADP crystals 
results in fracturing. Cracking of crystals 


cemented to a support owing to differential 
thermal expansion will be discussed in Sections 
8.6.4 and 8.6.5. 

Electrical Properties of ADP 

The electrical characteristics of particular 
concern to this chapter are the volume resistiv¬ 
ity and the surface resistivity. Of the two, the 
volume resistivity overshadows the latter in 
importance since the surface resistance of ADP 
is about fifteen times as great as its volume 
resistance. The surface resistance does not seem 
to be materially affected by humidity unless the 
latter attains a very high value. However, in 
handling ADP crystals, the surface conductivity 
must be considered, especially from the stand¬ 
point of any increase owing to contamination. 
The remarks in the introduction of this chapter 
on handling precautions are applicable in this 
connection. 

The volume-resistivity characteristic of ADP 
is quite sensitive to the existence of impurities 
in the growing solution, the most important 
impurity being sulphate. The resistivity of ADP 
as a function of the sulphate content of the 
saturated solution in which the mother crystal 
is grown has been determined at BTL'^ and is 
shown by a graph in Figure 5. In the specifica¬ 
tions outlined in Section 8.5.7, it will be noted 
that ADP crystals are graded on the basis of 
their volume resistivity. 

The volume resistivity for a Z-cut ADP crys¬ 
tal as a function of temperature, as measured 
by Johnson and Briggs,® is reproduced in Fig¬ 
ure 6 . Similar data have been obtained by other 
workers. Typical resistance values between the 
electrode faces of a clean ADP crystal 1 cm 
square in area and 2 mm thick, in dry air, as 
given by the Brush Development Company® are: 


Temperature 
in degrees C 

Resistance 
in megohms 


25 

1,500 


35 

1,000 


46 

500 


75 

100 


100 

17 



The leakage conduction in ADP crystals is 
therefore found to be quite different from that 










274 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


in RS. In an RS crystal the conductivity is pri¬ 
marily a surface phenomenon and depends on 
the relative humidity of its environment. Even 
when immersed in a water-free liquid, RS will 
furnish its own moisture from its water of 
crystallization to bring about lower surface re¬ 
sistance and perhaps, under operating condi¬ 
tions, eventual electrical breakdown. Since ADP 
crystals contain no water of crystallization to 
escape, their surface resistance remains high 


tion of the surface insulation in ADP must be 
well above 100 C. Even storage at 100 C for 
6 months does not produce any permanently 
adverse effect on surface leakage.® 


Storage Conditions for Crystals 

In the use of RS, it is necessary to control 
the humidity for best results. Only by so doing 



Figure 5. The electrical resistivity of ADP crystals as a function of the sulphate content of the growing- 
solution. 


even while immersed in oil at high tempera¬ 
tures. With ADP, however, high temperatures 
may produce a loss of ammonia from the sur¬ 
face and thereby result in a surface coating of 
phosphoric acid. This acid coating would natu¬ 
rally result in high electrical leakage. The tem¬ 
peratures necessary for permanent deteriora- 


is it possible to attain minimum electrical leak¬ 
age and minimum transient effects. It is also 
important to avoid dehydration of the crystal 
surface since this could lead to a high-voltage 
drop in the dehydrated layer. At a relative 
humidity below about 35 per cent, RS crystals 
lose moisture and dehydrate, while above 85 per 






























































PREPARATION OF INDIVIDUAL CRYSTALS 


275 


cent relative humidity, moisture collects on the 
surfaces and the crystals dissolve. The safe 
upper and lower limits of relative humidity as 



Figure 6. Electrical resistivity of ADP crystals 
as a function of temperature. 


a function of temperature for the proper stor¬ 
age of RS crystals are shown graphically in 
Figure 1. In general, it is better to keep the 
relative humidity for crystals stored in air near 


the lower limit shown on the graph. This can 
be done at all temperatures by enclosing the 
crystals in a hermetically sealed box, together 
with a mixture of hydrated and dehydrated RS 
in powdered form. If sufficient care is exercised 
to remove adsorbed water from all the hydrated 
salt concerned, then dehydrated salt need not 
be mixed with the hydrated RS for control pur¬ 
poses. It is usually desirable, however, to use a 
small quantity of dehydrated salt to insure the 
removal of excess moisture. In regard to the 
amount of RS required, the following rule may 
be used: allow 1 per cent of the enclosed volume 
to be control salt. This allows a very high factor 
of safety. 

If RS crystals are to be surrounded by oil, 
any moisture on them should be removed by 
placing them in a vacuum chamber for perhaps 
5 or 10 minutes. The length of time in which RS 
may be evacuated without serious dehydration 
is markedly dependent on its surface texture. 
Likewise, the moisture and air should be re¬ 
moved from the oil before immersion of the 
crystals. This point is treated at some length, 
and complete equipment for dehydration is dis¬ 
cussed, in Section 8.8.9. The remarks in this 
paragraph are also applicable to ADP crystals. 

As regards a safe upper temperature limit 
for the storage of RS crystals, the value of 50 
C or 122 F is suggested. 

The storage of ADP crystals does not present 
any special problems. The relative humidity 
should not exceed 90 per cent, however, nor 
should the temperature be appreciably over 100 
C. At temperatures higher than 100 C, it would 
probably be desirable or even necessary to have 
an atmosphere of ammonia vapor to prevent de¬ 
composition of the crystal-surface layers if 
storage were contemplated for an appreciable 
length of time. 


PREPARATION OF INDIVIDUAL 
CRYSTALS 

® ^ ^ Orientation of RS Bars 

The appearance of a synthetically grown RS 
bar intended for X-cut crystals is shown dia- 
grammatically in Figure 8. The location of 








































276 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


X-cut and Y-cut shapes is also shown in relation 
to the principal surfaces of the mother crystal. 
In order to obtain crystals of the desired orien¬ 
tation, it is necessary to start with accurate 
reference planes on the mother bar. It has been 
found that the two larger sloping surfaces of an 
RS bar do constitute satisfactory reference 
planes. A practical procedure is to design and 
build a jig on which these two reference planes 


Rough-Cutting Crystals from RS Bars 

An RS bar intended for the production of X- 
cut crystals is shown diagrammatically in Fig¬ 
ure 8. A schedule of the necessary steps for 
roughing out either X-cut or Y-cut crystals as 
desired, can easily be prepared from an inspec¬ 
tion of this diagram. For the production of X- 
cut crystals the base reference surface of the 



Figure 7. A Rochelle salt mother crystal whose seed bar is being milled off to form a reference surface. 


may rest in the correct position while the large 
base of the crystal which contains the seed bar 
is surfaced by means of a milling cutter. Figure 
7 illustrates the position of a mother crystal of 
RS on such a fixture during this particular sur¬ 
facing operation. In addition to the large base 
surface thus obtained, a second reference plane 
is desirable which will be perpendicular to the 
base plane and parallel to the long dimension of 
the bar. This may be produced by a second mill¬ 
ing operation while the bar is still in the same 
jig. 


bar is held in a vertical position against a guide 
and thin slabs are obtained by cutting with a 
band saw, as discussed in Section 8.4. In this 
manner, the entire bar is sliced into several 
large thin slabs which are then ready for fur¬ 
ther subdivision into individual rough crystal 
shapes. 

The slabs are next sawed obliquely into strips 
with the aid of a 45-degree guide. The same type 
band saw may be used as for the above slab cuts. 

The strips may now be sawed into rough-cut 
crystal shapes by again using a band saw and 






PREPARATION OF INDIVIDUAL CRYSTALS 


277 


the necessary guides. In every case it is desir¬ 
able to have the rough-cut crystals oversize by 
about 0.040 in. in all three dimensions to allow 
for later finishing operations. 

If Y-cut crystals are desired, the slabbing of 
the bar shown in Figure 8 would be done by 
making longitudinal cuts along a plane per¬ 
pendicular to the large reference surface. The 
further processing of the slab into rough-cut 
crystal shapes would be done in a manner 
analogous to that already outlined for X-cut 


be adequate. For production work, milling 
equipment involving rotary tables of the type 
illustrated in Figure 26 will probably be much 
more economical. With four milling heads 
around a turntable the cutters may be adjusted 
so that both a coarse and a fine-finish cut will 
be taken during each half revolution. 

Dry-grinding processes have not proved sat¬ 
isfactory for finishing RS crystals. 

8.3.t Orientation of ADP Bars 



Figure 8. Diagram of the position of 45° X-cut 
and 45° Y-cut Rochelle salt crystals in the 
mother bar. 

crystals. For the economical production of Y 
cuts it would be preferable to grow the mother 
bar originally from a seed bar so oriented that 
the Y cut could be obtained by exactly the same 
steps as have been outlined above in detail for 
X cuts. 


8.3.3 Surface Finishing RS Crystals 

The Brush Development Company has had 
very extensive experience over a period of many 
years in the processing of RS crystals. In their 
current production work on surface-finishing 
RS crystals they rely exclusively on milling 
processes. The actual description of their mill¬ 
ing equipment will be presented in Section 8.4.5. 
The milling equipment discussed in Section 
8.4.6 for ADP crystals is also applicable to RS. 

The final choice of equipment for milling RS 
will depend on the volume of work to be done. 
Where relatively few crystals are being proc¬ 
essed, as in an experimental laboratory, the ap¬ 
paratus described in Section 8.4.6 will probably 


Figure 9. Sketch indicating the location of a 
45° Z-cut plate and a capped seed plate within 
the mother ADP crystal. 

visible at the central section of the bar and on 
either side of it are the two white, pyramidal 
seed caps. The composition of the solution into 
which the seed plate is first introduced is such 
as to force these caps to form very rapidly. The 
rapid initial growth, though economical, is re¬ 
sponsible for the lack of clarity in this region 
of the bar. The usable portion of the bar is the 
clear area lying between the extreme tips of 
these seed caps and the pyramidal faces at 
either end. 

The location within the mother bar of the 45- 
degree Z-cut ADP plates, the only type cut used 
in transducers, is indicated in Figure 9. In order 
to obtain these crystals in rough form from the 
original bar, one must first establish reference 
planes. The long prism faces of the bar are not 
satisfactory for this purpose since they have 
an appreciable taper which is unavoidably in¬ 
curred during growth. The pyramidal end faces 
of the crystal, however, are sufficiently flat and 


A photograph of an almost perfect ADP 
mother bar is reproduced in Figure 4. Its size 
can be appreciated by comparing it with the 
woman’s hand which rests upon it. This par¬ 
ticular bar was grown in a reciprocating radial 
crystallizer. The clear flat seed plate is plainly 












278 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 



the pyramidal angles are quite constant. The 
pyramidal faces each make an angle of 44° 50' 
with the longitudinal optic axis of the bar. 

In processing a bar the central seed plate 
region is first removed by means of the abra¬ 
sive cutoff wheel described in Section 8.4.3, 


Figure 10. Orientation of the pyramidal end of 
an ADP mother crystal in a special jig for 
grinding reference surface. (Naval Research 
Laboratory.) 

leaving the two clear end portions. Reference 
surfaces are established on these two end por¬ 
tions by either one of two procedures. The 
method currently used at NRL employs a spe¬ 
cial jig in which the pyramidal end of the bar 
may rest, as illustrated in Figure 10. While 
being held firmly by hand in a vertical position 
in this jig, one corner of the bar is ground off 
with a vertical belt sander at an angle of 45 de¬ 
grees to a depth sufficient for a reference plane. 
If desired the other three corners of the bar 
may be ground off similarly so that a square bar 
results. This square bar is then sliced directly 
into 45-degree Z-cut plates with an abrasive cut¬ 
off wheel as described in Section 8.4.3. 


An optical method of providing proper orien¬ 
tation of the original bar has been used at BTL. 
This method of orientation will be made clear 
by careful examination of the instrument shown 
in Figure 11, and referred to as a “reflectoria- 
scope.” An automobile headlight lamp at the 
left supplies light which is collimated in the 
direction of the longitudinal axis by means of a 
lens. This parallel beam falls on two adjacent 
pyramid faces which then reflect separate por¬ 
tions of the beam through each of two lenses 
so that images of the lamp filament are focused 
on two white screens at right angles to each 
other. If the crystal is turned slightly these 
images move on their screens. By adjusting the 
crystal properly both images can be caused to 
center on cross lines ruled on the screen. When 
these images are so centered the crystallo- 


Figure 11. Reflectoriascope for orienting ADP 
mother crystals, preparatory to grinding refer¬ 
ence surfaces for locating 45° Z-cut plates. (Bell 
Telephone Laboratories.) 

graphic axes are parallel with the edges of a 
mounting board which is under the crystal. If 
the crystal is secured in this position it can be 
accurately cut while using the mounting-board 
surfaces as planes of reference. The mother bar 
is attached to the bakelite plate by means of a 
thermosetting plastic cement which sets at 
room temperature. This Norace cement, when 
first mixed, has the consistency of putty so that 
the position of the crystal can be readily ad¬ 
justed when on the reflectoriascope and yet re¬ 
tain its correct position by subsequently taking 







PREPARATION OF INDIVIDUAL CRYSTALS 


279 


a permanent set. For further directions on the 
use of this cement see Section 8.6.8, 

The bakelite plate containing the properly 
mounted ADP bar ends is now held against a 
45-degree angle block on the guide table of a 
belt Sander. The reference surface is ground as 
illustrated in the photograph in Figure 20. The 
photograph shows the left half of the crystal 
being sanded, the right half already having 
been ground down. When the crystal block is 
later sliced into Z-cut plates of the desired thick¬ 
ness, each slice will contain a portion of this 
reference surface. 

Where the half-bar is held vertically on a spe¬ 
cial jig which fits the pyramidal end faces, while 
the reference surfaces are ground by a vertical 
belt Sander, the maintenance of correct orienta¬ 
tion of the ADP bars depends largely on the 
skill of the operator. This procedure, which has 
been used and recommended at NRL, appar¬ 
ently works out very well for the currently 
available sizes of ADP mother bars. It appears 
that the usable part of an ADP bar could be¬ 
come so long that difficulty would be experi¬ 
enced in obtaining a correctly oriented and sat¬ 
isfactorily flat reference surface by this method. 
The optical orientation method employed at the 
BTL would seem to be preferable from the 
standpoint of maintaining accuracy of the ref¬ 
erence plane, particularly for grinding the us¬ 
able end portions of longer ADP bars. However, 
the optical method involves a cementing opera¬ 
tion which requires setting time and is undoubt¬ 
edly much slower. The additional accuracy 
would not justify the increased cost in many 
cases. 

While the dry-grinding method of producing 
reference surfaces is rapid and apparently 
quite satisfactory, it is also possible to use a 
liquid cooled abrasive cutoff wheel for this pur¬ 
pose. Such wheels are regularly used for the 
subsequent slicing of ADP bars, as discussed in 
Section 8.4.3, and would therefore be generally 
available. 

8.35 Roiigh-Ciittiiig Crystals from 
ADP Bars 

Following orientation and the grinding of 
reference surfaces on ADP bars, as just out¬ 


lined in Section 8.3.4, the bars are best cut into 
slices by means of wet abrasive cutoff wheels. 
This type of equipment is illustrated in Figure 



Figure 12. Laying out crystals before finish 
cutting to correct length and width. (Bell Tele¬ 
phone Laboratories.) 


25 and its use is discussed in some detail in Sec¬ 
tion 8.4.4. It will have been made clear from an 
inspection of the diagram of Figure 9 that 45- 



Figure 13. Grinding crystals to length and 
width dimensions on belt grinder. A fine grit belt 
is used for the finishing cut. (Bell Telephone 
Laboratories.) 

degree Z cuts are obtained by slicing the bar at 
right angles to its long axis. The thickness of 
the slices should be about 0.040 in. oversize in 







280 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


order to allow for later finishing cuts. To in¬ 
sure that these slices have parallel surfaces, 
they may be held next on a special vacuum 
chuck arranged parallel to the belt, and there¬ 
fore oriented at right angles to the bed, of a 
belt Sander shown in Figure 13. Then both elec¬ 
trode faces of the crystal plate are ground down 
to about 0.017 in. oversize and made parallel 
with each other. 

Where all four corners of the mother bar are 
ground to produce reference surfaces, so that a 
square cross section results, merely sawing 
these bars into slices yields rough 45-degree Z- 
cut crystal shapes. If only one side of the bar 
has been given a reference surface it may be de¬ 
sirable to lay out a cross section of the crystal 
shape desired by means of a parallelogram ar¬ 
rangement such as illustrated in Figure 12. 
This procedure permits an inspection of the 
crystal plate and enables the rough shape to be 
laid out so that faulty spots on the slice are 
avoided. The desired crystal shape may then be 
roughed out by placing the crystal slice on an 
appropriate jig and grinding it to the proper 
oversize dimensions with a belt sander. The 
illustration in Figure 13 will make this pro¬ 
cedure clear. 

Abrasive cutoff wheels furnish an alternative 
method of obtaining rough crystal shapes from 
the original slices. Preference here seems to lie 
with the individual and to depend also on the 
availability of equipment. Liquid coolants are 
always employed with these wheels when cut¬ 
ting ADP crystals. For some crystal sizes, 
economy considerations alone would necessitate 
the use of a sawing procedure. Since the use of 
band saws with ADP is not feasible, recourse is 
had to cutoff wheels of the abrasive type or to 
diamond wheels. 


Surface Finisliiiig ADP Crystals 

Surface finishing of ADP crystals may be ac¬ 
complished satisfactorily by any one of several 
techniques. A dry-grinding process has been de¬ 
veloped and used rather extensively at NRL, at 
BTL, and at the Hawthorne crystal-growing 
plant of the Western Electric Company. Milling 
processes which had been developed and used by 


the Brush Development Company over a period 
of many years for RS have also been applied to 
ADP. The Brush Development Company con¬ 
tinues to use milling equipment for this purpose 
and UCDWR has also employed a modified 
milling technique. Wet-grinding equipment has 
been adapted to finishing ADP surfaces at NRL, 
where use has been made of thin abrasive cutoff 
wheels bonded to steel disks for this purpose. 

To obtain correct length and width dimen¬ 
sions by means of a dry sander, a 120-grit belt 
is used and the chuck is arranged to employ 
stops as shown in Figure 13. A dry disk sander 
could be used but is probably not quite as de¬ 
sirable. In order to finish to the correct thick¬ 
ness the crystals are then transferred to a belt 
surface grinder such as illustrated in Figure 22 
and described in greater detail in Section 8.4.2. 
In this device, which was used at BTL,' a num¬ 
ber of crystals are held on the vacuum chuck at 
one time and are passed under a pulley carrying 
a fine abrasive belt. Equipment of this type is 
suitable for laboratory use or for small-scale 
production. For large-scale production, surface 
finishing may be done more economically with 
the type of equipment shown in Figure 26. 

At UCDWR the finishing of ADP crystals has 
consisted primarily in altering the dimensions 
of crystals which were previously furnished in 
a finished condition but whose dimensions were 
not appropriate for the application at hand. The 
type of milling equipment employed is illus¬ 
trated in Figure 27 and has been discussed in 
detail in Section 8.4.6. Its principal advantage 
lies in the ease with w^hich its simple sweep¬ 
cutting tool can be reconditioned. The larger 
and more expensive cutters, such as the spiral- 
end mills used by the Brush Development Com¬ 
pany, are more difficult to resharpen. In general, 
it may be said that milling operations on ADP 
place greater demands on the cutting tools and 
hence are less satisfactory and more difficult 
than in the case of RS. Since this is largely a 
matter of securing satisfactory cutting tools, 
more favorable alloys for this purpose may be 
eventually developed. If so, a milling process 
might still become the most satisfactory tech¬ 
nique for finishing ADP crystals. 

Since abrasive cutoff wheels employing a 
liquid coolant have proved entirely satisfactory 




PREPARATION OF INDIVIDUAL CRYSTALS 


281 


for slicing ADP, they would seem to be adapt¬ 
able also for the finer finishing operations. 
Actually this is the case and NRL has used the 
identical type of abrasive cutoff wheels for this 
purpose by bonding them to metal disks. A 
photograph of their equipment is shown in Fig¬ 
ure 23 and a detailed discussion will be found in 
Section 8.4.2. Accurate duplication of crystal di¬ 
mensions is facilitated by a micrometer adjust¬ 
ment and the use of a hydraulic-feed mecha¬ 
nism. 

Since any of the techniques mentioned above 
succeed in producing entirely acceptable results 
on ADP crystals, the choice between them has 
usually rested on individual preferences and on 
the availability of equipment at the several 
laboratories. Sometimes a choice can be made 
on the basis of the quantity of crystals being 
processed. For example, the sweep-type milling 
cutter used at UCDWR is not well suited to 
mass production although it possesses many 
advantages for the experimental laboratory. 
Wet-grinding equipment as developed up to the 
present time is likewise not well suited to mass 
production. On balance, it would seem that the 
dry-grinding equipment is most economical 
where great numbers of crystals are involved. 


^ ’ Spliced Crystals 

For low-frequency applications, it may be dif¬ 
ficult or impossible to secure single crystals of 
sufficient length to have a resonance in the de¬ 
sired region. However, it has been found that a 
sufficiently long crystal may be obtained by 
bonding together two or more smaller crystals. 
This bonding operation may be performed be¬ 
fore the original bars are cut into individual 
crystals. These spliced crystals can readily be 
detected visually and it is important to distin¬ 
guish them from those grown originally as 
single crystals. 

In fabricating RS crystals, melted RS may 
be used as a cement according to the directions 
in Section 8.6.9. Thus, the bonding layer has 
properties almost identical to that of the crystal 
itself. No special precaution is necessary when 
bonding spliced RS crystals to supporting 
structures by the application of the regular ce¬ 


menting techniques outlined later in Section 8.6. 

In commercially available Y-cut RS crystals 
the splice has appeared at an angle of 45 degrees 
when viewed on the electrode face as illustrated 
in Figure 14. By looking at the location of Y-cut 
crystals in the mother bar it will be seen that 
there is an economical advantage in bonding 
these crystals at the 15-degree angle. Voltage 



Figure 14. Spliced crystals. Left, a 45° bond; 
right, a 90° bond. 

tests have been made at UCDWR on several 
hundred such crystals having dimensions 
3xlxl^ in. It was found that only about 1 per 
cent resulted in voltage breakdown within the 
cemented joint when tested at 6,000 v. It seems, 
therefore, that spliced RS crystals are about as 
dependable from a voltage standpoint as those 
cut in a single piece from the mother bar. 

Large ADP crystals can also be formed by 
bonding two smaller crystals together, but with 
a thermoplastic cement such as Vinylseal. De¬ 
fects at the bond are readily detected visually, 
owing to the increased reflectivity in areas 
where the bond has failed. Since the cement 
softens at higher temperatures there may be 
applications where such crystals cannot be used. 
Specifically, the Cycle-Welding technique for 
bonding ADP crystals to rubber, described in 
Section 8.6.6, involves temperatures which are 
excessive for this type of fabricated crystal. The 
spliced ADP crystals which have been available 













282 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


commercially have had the splice through the 
center of the crystal and perpendicular to the 
long dimension as in Figure 14. Since these are 
45-degree Z cuts, it would seem that some 
economical advantage could also be gained by 
having the splice at an angle of 45 degrees. 


Electrodes 

Electrodes may be applied to crystals in any 
one of several ways. Each laboratory which 
makes use of piezoelectric crystals apparently 



Figure 15. Equivalent circuit of crystal and 
its electrodes when a dielectric cement layer 
intervenes. 

prefers its own adopted method to the exclusion 
of all others. This would seem to indicate that 
any one of several procedures may be entirely 
satisfactory. According to the Navy specifica¬ 
tions^ for ADP crystal plates, metal electrodes 
may be applied by any method, such as plating, 
foiling, sputtering, or depositing by evapora¬ 
tion, provided such electrodes pass certain elec¬ 
trical and mechanical tests enumerated in the 
specifications. These tests follow: 

5-2 The metal electrode shall be substantially uni¬ 
form in thickness over the entire electroded surface, 
and shall adhere to the crystal sui’face sufficiently well 
to meet the following adherence requirements: (a) 
After a crystal is passed once a distance of approxi¬ 
mately 6 inches over a cloth or felt surface saturated 
with carbon tetrachloride or other suitable solvent and 
uniformly exerting a force on the electrode sui'face of 
the crystal of 12 to 16 ounces, the electrode shall appear 
to be of uniform thickness over the entire surface as 


gauged by eye, and shall meet the electrical resistance 
requirement of paragraph 5-3. 

5-3 Electrical uniformity of the crystals shall be 
determined' by measuring the d-c resistance between 
both sets of diagonal corners of the electrode surface by 
means of blunt gold-plated contact probes. The re¬ 
sistance measured in this manner shall not exceed 20 
ohms in either diagonal for the crystal sizes listed in 
paragraph '4-1. The term “corner” is taken to mean a 
point approximately Vs" in from either of the two 
edges of the crystal forming the corner. 

The largest electrode surface listed in para¬ 
graph 4-1 referred to above is 1x1.1 in. 

When a metal foil is used as an electrode, a 
thin layer of adhesive intervenes between the 
foil and the crystal. This means that the elec¬ 
trode is electrically coupled to the crystal 
through a capacitance C ,. This capacitance is 
in series with the crystal capacitance C^, and 
it is apparent that C . must be large (i.e., the 
cement layer must be thin) compared to or 
an appreciable fraction of the available voltage 
drop may occur in the adhesive layer. The 
equivalent circuit is drawn in Figure 15. As¬ 
suming the dielectric constant of the adhesive 
to be about 3 or 4, the problem is not serious 
with Y-cut RS or Z-cut ADP where the dielec¬ 
tric constant of the crystalline material is about 
10 or 14. With X-cut RS, which may have a 
dielectric constant of several hundred, however, 
it may be difficult to make the adhesive layer 
thin enough, especially when very thin crystals 
are used as in bimorphs. A solution to this prob¬ 
lem is the application of a conducting layer di¬ 
rectly on the crystal by evaporating metal or by 
spraying a conducting solution. Marked in¬ 
crease in voltage sensitivity is claimed to result 
in the case of X-cut bimorphs with electrodes of 
this type. 

Evaporated Electrodes 

In order to achieve intimate contact with the 
crystal surface it has become the practice in a 
number of laboratories to evaporate metallic 
substances onto the electrode faces of the crys¬ 
tals. Owing to its general chemical inertness 
and the high quality of the low-resistance elec¬ 
trical contact readily obtainable by pressure, 
pure gold has been most widely chosen for this 
application. The gold may be applied either by 
evaporation or by cathode sputtering. The sput- 













PREPARATION OF INDIVIDUAL CRYSTALS 


283 


tering process is probably undesirable because 
of the high temperatures developed at the crys¬ 
tal surface unless extreme precautions are 
taken, and the greater difficulties involved in 
controlling the gas pressure and the thickness 
of the film. 

Gold may be readily evaporated by hanging 
small hairpin loops of 0.020-in. gold wire, spaced 


The practical application of evaporated gold 
to crystals presents primarily an engineering 
problem. The vacuum equipment should have 
sufficient capacity to attain a pressure of ap¬ 
proximately 10~^ mm of mercury in a very short 
time. Large valves in the vacuum line should 
enable the diffusion pump to be connected or 
disconnected at any time without waiting for 



Figure 16. Apparatus for the evaporation of gold electrodes. 


at regular intervals, on a straight 0.040-in. 
tungsten wire and heating the latter to incan¬ 
descence. A schematic arrangement for this 
process is shown in Figure 16. With the tung¬ 
sten wire at a distance of 3 or 4 in. from the 
crystals, a practical evaporation equipment 
should permit the electroding of 100 or more 
crystals in one operation. 


it to warm up in advance or to cool off before 
opening the system. These valves should be 
such as to give long periods of carefree opera¬ 
tion. To prevent the crystals from cracking due 
to uneven heating, it is important that the gold 
be evaporated within a matter of seconds. Actu¬ 
ally, it amounts to a flashing process. The thick¬ 
ness of the foil can be controlled most readily by 













284 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


measuring the amount of gold which is sus¬ 
pended on the tungsten wire, assuming that the 
temperature is raised to the point where all the 
gold evaporates from the tungsten for each op¬ 
eration. A similar process has been used even 
for RS crystals by BTL, but it is clear that they 
may be subjected to a vacuum for a very limited 
time only without dehydration and that any de- 



Figure 17. Arrangement for sandblasting crys¬ 
tal surfaces. (Naval Research Laboratory.) 

terioration which could result from exposure to 
high temperatures must be avoided. 

The cleaning of crystals prior to evaporation 
of the metal should be such as to leave them 
completely free from fat soluble substances. 
Carbon tetrachloride is an excellent solvent for 
this purpose. The crystals may be washed in 
carbon tetrachloride, or perhaps still better, a 
continuous degreasing apparatus may be set up 
in which carbon tetrachloride vapor condenses 
on the crystals. In some laboratories it is felt 
that better adhesion of the metal to ADP is ob¬ 
tained if the crystal surface is lightly sanded 
before being subjected to the evaporation proc¬ 
ess. The sandblasting equipment used for this 
purpose at NRL is shown in Figure 17. As a test 


for the satisfactory adhesion of gold to crystals, 
one should wipe the electrode with a cloth wet 
with carbon tetrachloride, as stated in the Navy 
specification quoted in Section 8.3.8. This test is 
based on the fact that unsatisfactory electrodes 
usually result from the presence of fatty ma¬ 
terial on the crystal surface and hence gold de¬ 
posited on top of fatty substances would be 
wiped off by rubbing with a cloth wet with a 
fat solvent. 

It will be found necessary to protect the edges 
of crystals during evaporation in order to main¬ 
tain their high d-c resistance. This may be 
achieved by covering the edges of all crystals 
with a metallic baffle during the evaporation 
process. An alternative procedure might consist 
of placing rubber spacers between each crystal 
and maintaining a slight compression so that it 
is impossible for the gold to penetrate to the 
edge faces of the crystals. These two sugges¬ 
tions for baffles are depicted in the drawing of 
Figure 16. At least one laboratory finds that 
more satisfactory adhesion of gold may be ob¬ 
tained by first evaporating a film of aluminum 
on the crystals. The aluminum gives a hard 
coating which is very difficult to remove, while 
the gold serves as an inert protective covering 
for the aluminum. This double evaporated layer 
can be applied conveniently in the apparatus of 
Figure 16 because the aluminum evaporates 
first upon heating the tungsten element and is 
followed later by the gold. 

Sprayed Electrodes 

A second method of electroding crystals, 
which has found rather wide application, con¬ 
sists of spraying a conducting suspension or a 
molten metal on the crystals. At least three 
different materials have been used for this pur¬ 
pose, namely, molten tin, metallic silver in sus¬ 
pension, and graphite in suspension. 

The silver suspension which has been used 
satisfactorily for some time at NRL for ADP 
crystals is made up according to the following 
formula: 


Powdered silver, DuPont V-9 250 g, 

MM cement, EC678 100 cc, 

MM cement, EC658 10 cc. 

Ethylene dichloride (C 2 H 2 CI 2 ) 300 cc. 









PREPARATION OF INDIVIDUAL CRYSTALS 


285 


where MM represents a product of the Minne¬ 
sota Mining and Manufacturing Company. The 
powdered silver is added to the solvent and 
shaken vigorously for 10 minutes. The two MM 
cements are then added and mixed thoroughly. 
The silver suspension should be strained 
through cheesecloth to remove all large parti¬ 
cles. The silver preparation is best applied to 
the crystals by spraying; with a little experi¬ 
ence, it is not difficult to obtain a uniform layer. 
The proper thickness is gauged by the desired 
electric resistance of the electrode. This prepa¬ 
ration deteriorates with time so that it is neces¬ 
sary to make up new batches about every 10 
days. It was found that the resistance of the 
electrode obtained increased with the age of 
the silver suspension. For a crystal 1 in. square, 
the d-c resistance measured across the diagonal 
of one electrode surface normally lies within the 
range 1 to 4 ohms. The Navy specifications 
quoted in Section 8.3.8 set up a maximum of 20 
ohms. 

For RS crystals NRL has made use of elec¬ 
trodes obtained by spraying molten tin. A spray 
gun which has proved satisfactory for this pur¬ 
pose is obtainable from the Alloy-Sprayer Com¬ 
pany of Detroit, Michigan. 

Graphite electrodes have been used for many 
years in the manufacture of X-cut RS bimorphs 
for air microphones and phonograph pickups. 
These electrodes seem to be quite satisfactory 
for such applications although they are not 
recommended for high-power devices. This 
power limitation may result from faulty adhe¬ 
sion to the crystal and a subsequent deteriora¬ 
tion in the electrode, perhaps caused by local¬ 
ized overheating. It is suggested that the aque¬ 
ous graphite suspension (Acheson 1008) be 
sprayed on the crystal with a type WO de Vil- 
biss spray gun at 50 psi pressure. It is cus¬ 
tomary to apply two coats with a 10- to 15-sec 
drying period between them. The thickness of 
the graphite layer on the crystal should be such 
that few or no pin holes will be observed when 
examination is made by looking through the 
surface toward a strong light. In any event, it 
will be found necessary to protect an electrode 
of this type. One method of so doing is to attach 
an additional tin foil over the graphite electrode 
by a cementing process as discussed in a later 


section. Another method consists of attaching 
a narrow strip of foil to serve as an electric lead, 
and then to cover the entire crystal with a 
waterproofing compound. 

Foil Electrodes 

Thin metal foils may be obtained in any one 
of a wide variety of elements and their alloys. 
The selection of the most desirable material for 
use as an electrode depends primarily on corro¬ 
sion resistance and softness. The foils most fre¬ 
quently used have been pure silver, nickel silver 
or German silver, and pure tin. 

When pure silver foil is used, it is customary 
to anneal it so that it will adhere well to the 
crystals. The annealing is done in the tempera¬ 
ture range 1000 to 1100 F, preferably in an elec¬ 
tric oven, while keeping each sheet separate in 
order to prevent sticking. Coin or sterling silver 
does not soften under this annealing treatment 
and therefore has been found unsatisfactory as 
a foil. Pure silver that has not been annealed 
may break at points of flexing. This is particu¬ 
larly the case where narrow tabs may be sub¬ 
jected to a sharp bending action a few times. 

A pure silver foil 0.0017 in. thick seems to 
offer the best mechanical advantages as re¬ 
gards ease of handling and of soldering. Any 
slight irregularity or waviness in the annealed 
silver foil may be removed by stroking the foil 
with a l^-in. round steel rod while the foil is 
held against a fiat glass plate. The principal 
advantage in using this heavier type of foil for 
an electrode is that it serves both as an electrode 
and as the foil wiring strip which is so often 
required with other types of electrode. The in¬ 
creased stiffness of the foil permits soldering 
directly to it and for this purpose an extension 
of the foil is always provided for use as a wiring 
tab. This type of foil electrode can be conven¬ 
iently used for either single crystals, in which 
each crystal has its own soldering tab, or it can 
be used in long strips to which perhaps a dozen 
crystals or more are cemented, the entire strip 
possessing a single tab for soldering. Illustra¬ 
tions of various type tabs are shown in the fig¬ 
ures accompanying several sections under Sec¬ 
tion 8.7. 

Where long strip foils are desired, they may 
be cut by hand or preferably with a shear of the 




286 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


Diacro type, which have been made available ture of 110 F and at 60 per cent relative hu- 
by the O’Neil-Irwin Manufacturing Company, midity for a period of 12 hours or longer in the 
Minneapolis. Foils intended for individual crys- case of RS crystals; for ADP crystals curing 
tals are best cut by means of a special die for at a much higher temperature is permissible. At 
each size. Standard dies are normally too ex- UCDWR, an 80-C oven was used for ADP crys- 
pensive where only limited quantities of a given tals. 

size of electrode are needed. To take care of the Tin foil has been used extensively for crystal 
latter case, advantage has been taken of the electrodes' at UCDWR. This material has the 



Figure 18. Stages in the application of tin foil electrodes: A. Picking up foil with camel’s-hair brush 
dipped in Acryloid cement. B. Inverting foil over crystal. C. Removal of the brush. D. Rubbing foil to re¬ 
move excess cement and to insure close contact with crystal. E. Trimming off excess foil with razor blade. 


rather novel means of punching thin foil which 
is discussed in detail in Section 8.4.7. 

It has been the custom at UCDWR to attach 
silver foils to ADP crystals with bakelite BC- 
6052 cement or with Vulcalock cement. These 
cements may be either brushed or sprayed, but 
the most uniform results are doubtless to be ob¬ 
tained by spraying. After the foils are attached 
to the crystals and are firmly pressed by a rub¬ 
bing operation, they are placed in a pneumatic 
press and subjected to a pressure of 25 to 40 psi. 
While in the press they are cured at a tempera- 


thickness of 0.000275 in. and is apparently pure 
tin. This foil is malleable and is so thin that the 
vapor from the solvent in the adhesive is able 
to escape through the foil. The fairly rapid 
escape of the solvent is an important considera¬ 
tion since it prevents the electrode from becom¬ 
ing too readily damaged during assembly oper¬ 
ations. For electrodes which have been attached 
sufficiently ruggedly for use in transducers, 
practically all solvent must have evaporated. 

The technique of attaching tin-foil electrodes 
is shown in a series of illustrations in Figure 















PREPARATION OF INDIVIDUAL CRYSTALS 


287 


18A, B, C, D, E. The foils should be cut over¬ 
size so that they will extend about i/g in. beyond 
the edge of the crystal surface on all sides. 
These foils are conveniently picked up with a 
camel’s-hair brush containing a small quantity 
of adhesive as shown in Figure 18A. While the 
foil adheres to the brush, it is turned over and 
laid on the crystal. The brush is then withdrawn 
from beneath the foil, taking care to hold the 
foil on the crystal with a finger, as illustrated in 
Figure 18C. Inasmuch as an extremely thin 
layer of adhesive is desired between the foil and 
the crystal, it is necessary to massage the foil 
with a piece of soft cloth covering a finger, as 
shown in Figure 18D. Stroking and rubbing the 
foil rather briskly in all directions should result 
in an extremely close attachment. The cloth will 
take on a black discoloration in the process. 
After these foils have been applied to both sides 
of a crystal, the excess foil around the periphery 
may be trimmed off with a razor blade in the 
manner illustrated in Figure 18E. It will be 
noted that a fingernail is used as guide in cut¬ 
ting off the excess foil. The excess cement 
around the edge of the crystal should now be 
removed by means of a cloth dampened in a 
suitable solvent, such as methyl ethyl ketone. 

A satisfactory adhesive for attaching tin foils 
to crystals is Acryloid B-7. The original cement, 
as it comes from the manufacturer, should be 
diluted in the ratio 1 part Acryloid to 4 parts 
ethyl acetate. Further details are given in Sec¬ 
tion 8.6.9 covering the use of this adhesive. 

It will be observed that the appearance of the 
two sides of commercial tin foil are unlike, one 
being dull and the other polished. It has been 
customary to attach the dull side of the foil to 
the crystal in order to obtain a better bond. 

Where crystals with tin-foil electrodes are 
used as part of an array, electric connections 
must be made by cementing to them more sub¬ 
stantial strip foils, as discussed more fully in 
Section 8.7.4 on wiring. 


Polarizing Crystals 

Technique 

The method of polarizing crystals used at 
UCDWR is clearly indicated in Figure 19. Other 


laboratories use similar devices. The crystals 
are placed individually in the holder illustrated 
and a slight horizontal force is exerted to insure 
good electrical contact. It is not necessary that 
the crystals themselves possess electrodes pro¬ 
vided the crystal holder has electrode surfaces 
sufficiently large to cover an appreciable area of 
the crystal surface. With the crystal in the po¬ 
sition shown in the figure, a sudden downward 
thrust is exerted on the top of the crystal by 
means of some blunt object. The eraser end of a 
pencil has proved entirely satisfactory. At the 
moment of application of the force, the needle 
of the indicating instrument will give a rapid 
deflection followed an instant later by a sharp 
kick of the needle in the opposite direction as 
the force is released. According to the direction 
of the deflection, an appropriate mark is then 
placed on one side or edge of the crystal to indi¬ 
cate its polarity. The definition which estab¬ 
lished which side of the crystal receives the 
polarity mark is quite arbitrary so that it is 
quite possible that the same crystal could be 
marked in different ways at various labora¬ 
tories. The definition stated in the Bureau of 
Ships specifications is given in the following 
section. 

The extent of the deflection depends on both 
the actual force applied to the crystal and the 
suddenness with which the force is applied. It 
also depends on the sensitivity of the polarizing 
equipment, a discussion of which will be found 
in Section 8.4.8. Although a mere flicker of the 
needle is sufficient to establish the polarity, it is 
convenient to have a somewhat larger deflec¬ 
tion. With a sensitive instrument one can read¬ 
ily distinguish between X-cut RS, Y-cut RS, 
and ADP crystals by the amount of the deflec¬ 
tion. The deflection obtained from crystals of 
each of these materials decreases in the order 
named. 

It might be thought that the deflection ob¬ 
tained in the polarization process might furnish 
a reliable indication of the piezoelectric activity 
of the crystal, provided that an arrangement 
could be devised whereby each crystal tested 
could be subjected to the same impulse. Some 
preliminary trials in this direction were made 
at UCDWR and several hundred crystals were 
subjected to activity tests. Although a number 



288 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


of inactive crystals were discovered, this type of 
test was discontinued, partly because the ap¬ 
paratus had not undergone sufficient develop¬ 
ment, but largely because the measurements 
have no particular meaning. The conditions un¬ 
der which crystals are ultimately used were not 
closely duplicated in the test equipment. To be 
reliable the measurements would have to be 
standardized and humidity and temperature 



Figure 19. Diagram of polarity indicating 
equipment used at UCDWR. 


conditions brought under control. As far as the 
detection of an occasional inactive crystal is 
concerned, these may be spotted during the 
polarizing process just described. 

Marking 

The following statement on marking the 
polarity of crystals is taken from the specifica¬ 
tions of the Bureau of Ships'’ [BuShips], to 
which reference has already been made. 

The polarity of each crystal plate shall be clearly 
designated by a mark on that electrode surface which 
becomes positive when pressure is applied in the direc¬ 
tion of the longest dimension of the crystal plate. The 
mark shall be placed at the upper right-hand corner of 
the face as it appears when the long dimension of the 
crystal is vertical. This shall not adversely affect the 
crystal plate characteristics, and shall be of a per¬ 
manent nature. 

It has been the practice at UCDWR to indi¬ 
cate the polarity of crystals by placing an 


arrow-shaped mark on the radiating face of 
each crystal. This method is convenient in that 
it permits visible inspection of crystal orienta¬ 
tion during assembly operations and in the 
completed transducer. It is essential that the 
ink used for marking shall be nonconductive. 
To date a commercial preparation with the 
trade name Dykem has been used. A color code, 
with green for ADP, red for Y-cut RS, and blue 
for X-cut RS has also proved to be a laboratory 
convenience. 


« » PROCESSING EQUIPMENT 

Even though it is possible to purchase crys¬ 
tals to the exact specifications required for a 
given application, it is not always feasible to do 
so for an experimental laboratory. This is par¬ 
ticularly true for the physical dimensions of a 
crystal which control its frequency and its 
capacitance. Accordingly, it is a great conven¬ 
ience to be able to vary these dimensions at will 
without experiencing the delays so often in¬ 
volved in obtaining delivery on special sizes. In 
fact, it has been the standard practice at 
UCDWR to maintain a fairly large inventory of 
crystals in a relatively small number of stand¬ 
ard sizes and to modify these to the exact di¬ 
mensions required for any particular trans¬ 
ducer under construction. It is therefore felt 
that the processing equipment and operations 
discussed in this chapter may find rather wide 
use for modifying the dimensions of crystals 
which were originally purchased in finished 
stock sizes. 

The viewpoint adopted in this section has 
been to present the various types of equipment 
currently in use for processing crystals at the 
various laboratories engaged in the construc¬ 
tion of transducers. Where possible, definite 
recommendations are made as to the type of 
equipment best suited for specific operations. 


‘ ^ Grinding RS 

Grinding processes are not currently em¬ 
ployed in large-scale production for finishing 
RS crystals and consequently no equipment for 











PROCESSING EQUIPMENT 


289 



this purpose will be discussed. Neither is a 
grinding method used in the production of ref¬ 
erence surfaces on the mother bar. 

For the final polishing of occasional crystals 
to be used in precise measurements, recourse 
may be had to very slow speed grinding or lap¬ 
ping processes, either wet or dry. The main 
point is to avoid overheating of the crystal. De¬ 
tailed references to articles on polishing of re¬ 
search specimens are given by Cady.^*^ 


« ’ " Grinding ADP 

It has already been indicated in an earlier 
section that the reference surfaces on ADP bars 
may be provided quite readily by a grinding op- 


Figure 20. Grinding reference surface on an 
ADP crystal block with a vertical belt sander. 
(Bell Telephone Laboratories.) 

eration such as illustrated in Figure 20. Unlike 
RS, ADP crystals can be ground very rapidly 
and satisfactorily with dry abrasive belts. Noth¬ 


Figure 21. A dry disk sander in the process of 
grinding an ADP crystal. (Naval Research 
Laboratory.) 

sure that it is thinner than the remainder of the 
belt. Since there is no abrasive at the joint heat 
is liable to be generated by it if too thick with 
a consequent cracking of crystals. It has been 
reported' that the 45-degree plane of the ADP 
bar, which is ground down in order to form a 
reference surface, can be ground easier than 
the prism faces. This favorable circumstance 
lessens the probability of cracking the crystal 
bar while grinding the 45-degree reference 
plane. Belt speed is not a critical factor. 

For rough-cutting ADP crystal shapes, the 
same sander is useful. This has been pointed out 
in Section 8.3.5 and the process of obtaining 
correct length and width dimensions is illus¬ 
trated in Figure 13. In addition, it may be neces¬ 
sary to rough-grind the thickness dimension, 
particularly if the crystal plate as sliced by the 
abrasive cutoff wheel does not possess parallel 


ing elaborate in the way of special equipment 
is required for this operation, ordinary commer¬ 
cial belt Sanders being quite acceptable. A ver¬ 
tical-type sander is convenient for this purpose. 
Best results appear to be obtained with a fairly 
coarse grit, one laboratory suggesting No. 40, 
another preferring No. 60. The joint which oc¬ 
curs where the two ends of the belt are ce¬ 
mented together should be inspected to make 








290 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


electrode faces. If so, provision for a satisfac¬ 
tory chuck must be made. At BTL," a vacuum 
chuck with a vertical surface was used to hold 
the crystals while they were ground against a 
vertical belt sander. The crystals should still be 
oversize to permit a finish cut to be taken, 15 to 
17 thousandths of an inch being sufficient. 

It is also possible to use a dry disk sander, 
either for grinding reference surfaces on ADP 
or for the rough shaping of crystal plates. The 



Figure 22. Surface grinder with abrasive belt 
for grinding crystal plates to size. (Bell Tele¬ 
phone Laboratories.) 


operation of a disk sander at NRL for grinding 
a reference surface is shown in Figure 21. 

Rough cut ADP crystals, about 0.017 in. 
thicker than that finally required, may be 
ground to their final thickness dimension with 
a belt sander. This grinding method for the fin¬ 
ish cut was used successfully at the Hawthorne 
Plant of the Western Electric Company during 
World War II. One type of belt sander adopted 
for this operation is shown in Figure 22. The 
crystals are held on a vacuum chuck which in 
turn rests on a traveling horizontal bed. This 
enables the crystals to be moved underneath the 


pulley which carries an abrasive belt. A ma¬ 
chine of this type may readily be improvised by 
altering small surface grinders so that a pulley 
replaces the ordinary abrasive wheel. The pro¬ 
vision of an idler pulley which is adjustable in 
angle insures the retention of the abrasive belt 
on the pulleys. In the grinder shown in the 
illustration, the hand wheel at the upper right, 
which raises and lowers the sanding mecha¬ 
nism, is graduated to 0.005 in. The vacuum 
chuck must have a very flat surface to avoid 
breakage of thin crystal plates and it has been 
found convenient to surface it with the same 
belt used for grinding the crystals. A fine belt, 
120 grit, is used for the finish cut. 

For production surfacing of crystals a 
double-head grinding unit such as illustrated in 
Figure 26 is very useful. Two Sanford grind¬ 
ing columns are shown mounted over a turn¬ 
table of the vacuum chuck type. Vacuum con¬ 
nections are made through blocks to only those 
regions of the table which are passing under 
the sanding-belt pulleys. Therefore, the crystals 
may be readily adjusted in position or removed 
from the table at any time except when the 
crystals are actually under the sanding belt. 
Two operators are required, one to feed each 
side of the turntable. This machine was used 
at the Western Electric Hawthorne works" and 
was capable of surfacing two sides of about 
3,000 crystal plates 1 in. square in an 8-hr pe¬ 
riod. A somewhat larger diameter table would 
permit a total of four belt sanders about its 
periphery, thus enabling a coarse and a fine 
finish cut to be taken on one side of each crystal 
during a half revolution. With an operator to 
turn the crystal over, the other surface is fin¬ 
ished on the second half of the revolution. 

Since some hazard, as well as annoyance, is 
caused by the ADP dust liberated while sand¬ 
ing, it is necessary to install a dust collector on 
dry-sanding equipment. One convenient way of 
doing this is to attach a commercial dust collect¬ 
ing unit to each dry sander. In the method used 
at NRL, the dust is drawn into the top of a large 
tank in which the coarse granules may settle 
and be salvaged; the finer particles are drawn 
through a water spray in the exhaust system. 

The final lapping of rough-cut ADP crystals 
may also be done on a wet sander. To date, only 








PROCESSING EQUIPMENT 


291 


disk Sanders have been employed for this pur¬ 
pose. The NRL equipment is photographically 
illustrated in Figure 23. A 120-grit silicon car¬ 
bide disk, 0.060 in. thick, has been recommended 
by NRL. The thin silicon carbide disk has been 
attached to the steel backing plate of the sander 
with Vulcalock cement, then cured for 3 hr at 
300 F in a press. 



Figure 23. A wet disk sander for the fine 
finishing of ADP crystals. (Naval Research 
Laboratory.) 


In this operation it is desirable to have the 
cooling liquid strike against the center of the 
disk from beneath the work table. Water and 
propylene glycol in equal parts have been sug¬ 
gested at NRL as a satisfactory solution for 
cooling. The solution is recirculated in the 
equipment continuously during operation and 
need be changed but once a month. Ethylene 
glycol would also be satisfactory were it not for 
its toxicity. Some laboratories'^ have found that 
a saturated aqueous solution of ADP is prefer¬ 
able to either of the above glycols. After grind¬ 
ing, the coolant solution may be removed from 
the crystals by immersing them in carbon tetra¬ 
chloride. 

Sawing RS 

All the rough cutting of RS bars into crystal 
plates may be done with band saws. This in¬ 


cludes the cutting of the mother bar into long 
thin slabs, the slabs into strips, and the strips 
into the final rough crystal shapes, all as out¬ 
lined in Section 8.3.2. A linear cutting speed of 
approximately 3,600 fpm has been found to be 
satisfactory, but it is not a critical figure. 
Nickel-steel blades have been found to give good 
service. Saw blades in use currently at UCDWR 
have ten teeth to the inch and were originally 
designed for wood cutting. New blades as pur¬ 
chased were usually found unsatisfactory for 
sawing crystals. Besides being insufficiently 
sharp, the rake and clearance angles and the set 
of the teeth did not have optimum values for 
this application. 

A relatively thin saw, about 0.02 in. or 
slightly less, is perhaps the best. The set of the 
teeth should result in an overall thickness about 
twice that of the blade material. All points on 



Figure 24. Band saw. Left: Face view showing 
the zero rake angle and the 30° clearance angle. 
Right: Edge view showing the set and the angle 
of filing the teeth. 

the saw should be set, with alternate points 
being set in opposite directions. Satisfactory 
shape and set of the saw teeth are illustrated in 
Figure 24. A zero rake angle appears to be de¬ 
sirable but not essential in the case of RS. Al- 








292 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


though the clearance angle is not critical, a 30- 
degree angle has been chosen at UCDWR, pri¬ 
marily for convenience in sharpening the teeth 
with a three-cornered file. A very fine file is 
recommended. Assuming that the saw is 
clamped horizontally, the correct file stroke for 
sharpening is to have the cutting edge of the 
file vertical and at right angles to the plane of 
the blade for all teeth. The sharpness of each 
tooth at the cutting point is of paramount im¬ 
portance. 

The Brush Development Company, which has 
processed RS crystals for many years, has spent 
a great deal of time in developing sawing tech¬ 
niques.^^ Their methods differ somewhat from 
the above, but mostly in detail only. Inasmuch 
as they regard their exact techniques as trade 
secrets, it is not possible to report in greater 
detail on their current methods. 

In rough-cutting RS, it is recommended that 
the crystals be made about 0.030-in. oversize in 
all dimensions in order to allow for finishing 
operations. To avoid chipping, it is extremely 
important to have the saw very sharp at all 
times. Chips are most likely to occur at that 
edge where the saw teeth leave the crystal. 
When sawing thin slabs or strips into individual 
crystals, it has been found at UCDWR that 
chipping of the bottom crystal face can be 
avoided to some extent by a preliminary scor¬ 
ing. This scoring is brought about by a back¬ 
ward rotation of the sample through 90 degrees, 
sawing a shallow groove in the bottom face, 
then turning the crystal back to its original 
position and completing the cut by sawing 
through the groove. 


« ^ " Sawing ADP 

It is so much more difficult to saw ADP crys¬ 
tals than RS crystals that an ordinary band 
saw has been found unsatisfactory in general. 
However, some use has been made of a band saw 
at UCDWR for cutting previously finished 
crystal plates in half, and with fair success. For 
sawing ADP, it is strongly recommended that 
the blades have the appearance of the illustra¬ 
tions in Figure 24. Frequent filing of the blades 
will be found essential. The remarks on chip¬ 


ping which appear in Section 8.4.3 are particu¬ 
larly applicable when sawing ADP plates. It is 
standard practice to saw ADP with a thin sili¬ 
con carbide wheel although diamond cutoff 
wheels have also been used. A satisfactory size 
and grade of silicon carbide disk according to 
experience at NRL is a 12-in. disk, 0.060 in. 
thick, with 120 grit. With the 12-in. silicon car¬ 
bide disk, a speed of 3,400 rpm was reported 
to be satisfactory. 

The equipment used at NRL is illustrated in 
the photograph of Figure 25, in the process of 



Figure 25. A wet abrasive cutoff wheel in the 
process of slicing an ADP crystal. (Naval Re¬ 
search Laboratory.) 


slicing an ADP bar. This apparatus is a Felker 
Model 120. An important feature for this type 
of work is the hydraulic-feed mechanism which 
furnishes a readily adjustable and uniform 
cutting pressure. The liquid coolant strikes 
across the radius of the disk on both sides. NRL 
suggests a cooling solution made up of 3 parts 
water and 1 part propylene glycol for this saw¬ 
ing operation. After cutting, the crystals are 
placed in trays containing carbon tetrachloride 
in order to remove the cooling liquid. 

In sawing through thick bars, it is essential 
that the cutting disk run true. Consequently, 
most disks will need to be trued following in¬ 
stallation. It is also essential that the disk have 
some taper from the edge inward toward the 
center to avoid binding. If disks are not avail- 







PROCESSING EQUIPMENT 


293 


able in this form, it will be found necessary to 
provide taper, especially for deep cuts. 

The experience of BTL on sawing ADP crys¬ 
tals is quoted from their report" as follows : 

The saws are abrasive cutoff wheels such as are used 
on metal. We have not found the grit size to be critical 
but prefer grits between 60 and 100. Silicon carbide or 
aluminum oxide seem to be equally satisfactory, the 
binder has not been found to be critical. 

Our work has shown that relatively slow wheel speeds 
are preferable to fast, partly because the machines in 


tremely little cross motion. Saw blades that have been 
used for a considerable time tend to become thinner 
near the edge than near the center. This causes a 
wedging action that not only makes the saw turn hard 
but also cracks thin slices. It can be eliminated by 
undercutting the saw faces with a diamond in a lathe. 
It has been found to be good practice to undercut 8" 
saws of i/io" thickness by about .007" on each side 
fi'om a point inside the periphery right to the 
pressure plates. 

A continuous stream of cooling fluid must flow to 
each side of the saw. At first small gear pumps were 



Figure 26. Production surfacing machine for crystal plates. For Rochelle salt, spiral milling cutters 
may be used; for ADP, sanding belts are used. 


which the saws are used are prone to build up vibra¬ 
tions at higher speeds. The machines were F'elker Di- 
met models No. IIB and No. 80, as used in quartz cut¬ 
ting. They were operated at about 1600 rpm with 8 inch 
saws although occasionally a 12" or 16" saw was used 
at the same speed for exceptionally large crystals. 

The saw blades were trued and the surface roughened 
simultaneously by using a straight knurl as a dressing 
tool, turning the saw slowly by hand so that the knurl 
crumbled the saw edge, leaving a fine roughness. Saws 
so treated cut several times as fast as those that were 
trued with a diamond. Sawing machines must be very 
carefully lined up so that as the saw moves through the 
crystal it moves quite accurately in a plane with ex- 


used for this purpose, but later they were replaced by 
centrifugal pumps which dipped in the solution. This 
obviates the priming problem. The centrifugal pump 
has no packing glands and no bearings that contact 
the fluid, the motor being high above the fluid. 

Besides saturated ADP solution, which has been 
found to be the most satisfactory saw coolant, mix¬ 
tures of propylene glycol and saturated ADP solution 
have also been used. Light oils like kerosene have also 
been tried but in every case the straight saturated ADP 
solution is preferred. 

In sawing large crystals it is very important to have 
the sawing fluid at the same temperature as the crystal, 
within about 2°C. The best arrangement is to maintain 










294 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


the whole cutting shop at a constant temperature 24 
hours of the day, and to cut no crystal until it has been 
in this room for several hours. This is particularly true 
in the winter when the day to night temperature varies 
widely if the plant runs only one shift per day. For a 
while all crystals were stored in an oven thermo¬ 
statically controlled at SOX. The saw coolant was also 
controlled at 30°C. 


« ‘" Milling RS 

The final surfacing- operations on RS crystals 
have been done with milling cutters. These cut¬ 
ters may be either spiral face milling cutters or 
spiral end mills. Spiral end mills with four 
flutes, a diameter of % in., and an operating 
speed of 5,000 rpm or more are suggested for 
this purpose. Sharp cutting edges are essential 
and a rather large clearance angle is recom¬ 
mended. 

Any one of several arrangements for milling 
should give equally acceptable results. Much de¬ 
pends on the actual quantity of the material 
being processed. For an experimental labora¬ 
tory an ordinary plain milling machine would 
be sufficient, but for production work it is ad¬ 
vantageous to install more elaborate equipment. 
The photograph in Figure 26 shows a type of 
small production machine with interesting pos¬ 
sibilities in this direction. This particular ma¬ 
chine is equipped with belt sanders but it is a 
simple matter to design milling heads for this 
type of equipment, as has been done by the 
Brush Development Company. The rotary table 
turns continuously at a speed which enables 
crystals to be placed on the table by one opera¬ 
tor and removed from it by another operator 
following the milling cut. By having two cutters 
in tandem, both a coarse and finish cut may be 
taken in sequence in the same operation. In fact, 
the process may be speeded up still further by 
having four milling cutters work on the same 
operating table. In this way one side of a crystal 
may be given a coarse and a finish cut while the 
table rotates 180 degrees, after which an opera¬ 
tor turns the crystal over so that the second 
surface is finished during the second 180 de¬ 
grees rotation. The crystals are held tightly on 
this rotating table by a vacuum chuck arrange¬ 
ment. To facilitate addition and removal of the 


crystals, only that part of the table in the im¬ 
mediate vicinity of the milling cutters is con¬ 
nected to the vacuum line. 

The milling equipment used at UCDWR for 
ADP, which employs a simple sweep cutting 
tool, functions just as satisfactorily for RS. A 
complete description occurs in Section 8.4.6. 


Milling ADP 

The milling of ADP crystals is a much more 
difficult task than the milling of RS. As a conse¬ 
quence, the milling cutters quickly become dull 



Figure 27. Vertical milling machine or jig- 
borer as used at UCDWR for finishing either 
ADP or Rochelle salt crystals. Note the support 
at the edges and sides of the crystals, also the 
use of a vacuum chuck. 

and frequent attention is required to maintain 
them in a sharp condition. Some of the harder 
cutting alloys are valuable for this application, 
but the ideal solution is yet to be found. To the 
writer’s knowledge, the Brush Development 
Company continues to surface ADP crystals 
with spiral milling cutters of the type discussed 


























PROCESSING EQUIPMENT 


295 


in Section 8.4.5. Their successful work with 
milling cutters on RS probably accounts for 
their adaptation of the same method to ADP. 

The task of maintaining sharp milling cutters 
for ADP crystals can be simplified if one adopts 
a vertical milling machine, commonly referred 




Figure 28. Position of stellite tool in vertical 
milling- head. Above: Side view showing angle 
made by tool shank with respect to the horizontal 
plane and also the cutting angle with respect to 
the horizontal crystal surface. Below: Edge view 
showing the zero rake angle and the 55° clear¬ 
ance angle of the stellite cutter. 


to as a jig borer, since it uses only a single¬ 
sweep cutting bit. Such a milling machine has 
been in use for some time at UCDWR. This 
type of equipment is shown in Figure 27. It will 
be noted that the stellite bit is inserted in the 
tool holder at an angle of 45 degrees. The crys¬ 


tals are placed on a traveling milling machine 
bed and held fast by means of a vacuum chuck. 
Rectangular bars of steel, also held on the table 
by means of the vacuum, are used to give sup¬ 
port to the crystals in the direction of motion of 
the cutter. The actual angular position of the 
cutter can be observed from the two drawings 
in Figure 28 which show vertical cross sections. 
The bottom illustration indicates that the clear¬ 
ance angle should be about 55 degrees and the 
angle of rake should be zero. The top illustra¬ 
tion of Figure 28 shows that the cutting edge 
should make an angle of approximately 10 de¬ 
grees with respect to the top surface of the crys¬ 
tals being cut. A satisfactory speed is approxi¬ 
mately 4,200 rpm. It will be found necessary to 
resharpen these cutters at fairly frequent in¬ 
tervals, but owing to their simplicity this is a 
very easy operation. The sharpening may be 
done on a disk sander having a very fine grade 
of Carborundum paper, grit No. 320A being 
satisfactory. The particular stellite bits used 
had a diameter of in. and a length of about 
2 in. Care must be exercised to avoid checking 
of stellite by heating during sharpening. 

With the sweep cutting tool just described, 
a rough cut of 15 to 20 thousandths of an inch 
could be taken from an ADP crystal each time 
the cutter traversed the block of crystals. For 
finishing cuts, 1 or 2 thousandths of an inch 
was satisfactory. To avoid chipping, the tool 
had to be sharp. 


* ‘ ' Die Cutting of Foils 

Where foil stampings are required in large 
numbers, it would probably be most economical 
to produce them by means of a standard-type 
die in a power press. In an experimental lab¬ 
oratory, however, there are occasions where 
only a few or perhaps a few hundred foils of 
a given size are needed. A simplified die-cutting 
process which is well adapted to small produc¬ 
tion has been used at UCDWR for some time 
and will now be described. 

An exact profile of the desired shape and size 
of the foil is marked off on a piece of thin steel 
stock and then machined or ground accurately 
to dimensions. This thin piece of steel is then 



































296 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


soldered to a large supporting plate as shown 
in Figure 29 and constitutes the male part of 
the die. The thickness of the steel stock is indi¬ 
cated in the figure as 0.05 in., but this dimen¬ 
sion is not critical and could be as thin as %■_> in. 
When machining grades of steel are used, the 
die should be hardened. It has been found, how- 



Figure 29. The die used for blanking thin foil 
electrodes. 


ever, that dies of this type may be conveniently 
ground from a very heavy grade of hacksaw 
blade. The base to which the thin steel die is 
soldered should have an area five or six times 
greater than the die. It is important that the 
edges of the die be perpendicular to its face and 
that the corners be very sharp. 



Figure 30. Blanking electrodes from silver foil 
with die shown in Figure 29. 

In stamping out silver foil with this die a 
simple arbor press may be employed. The die is 
held on the ram of the press and the silver-foil 
stock is placed on top of a thick sheet of rubber 
which rests on the bed of the press. The sheet 
of silver foil used should always have margins 
Vs to in. larger than the blanking die. The 
rubber should be Vo to 1 in. thick and should 


have a Shore durometer hardness test of 50 
to 70. Trials using different hardnesses of rub¬ 
ber should be helpful. When the ram of the 
arbor press is brought down, as depicted in 
Figure 30, the die is forced through the silver 
foil into the rubber. In this manner a very 
clean-cut replica of the original die may be cut 
from the foil. 

These dies should be capable of cutting sev¬ 
eral hundred blanks without resharpening. The 
blanking die may be sharpened by grinding 
down about 0.002 in. on its flat surface, care 
being taken to see that the corners are left as 
sharp as possible. 

This method of blanking thin sheets of any 
of the softer metals may find a wide variety 
of applications in an experimental laboratory. 
While production is not particularly rapid when 
a hand-operated arbor press is used, there is 
no reason why the process could not be made at 
least semiautomatic in a power-operated press, 
perhaps of the pneumatic type. 


^ ‘ ^ Polarizing Crystals 

The technique for polarizing and marking 
piezoelectric crystals has been discussed in Sec¬ 
tion 8.3.9. The actual indicating equipment 
would be expected to vary in detail from one 
laboratory to another. Since only a qualitative 
indication of polarity is sought and not a quanti¬ 
tative deflection, no great demand is placed 
upon the instrument. However, the deflections 
should be great enough to avoid undue incon¬ 
venience in reading them and to give unmis¬ 
takable indications. 

The electric circuit diagram for the equip¬ 
ment used at UCDWR is shown in Figure 31. 
It operates from a 110-v a-c line and gives very 
satisfactory deflections for the crystal sizes 
used in underwater sound transducers. Since 
the circuit elements are shown sufficiently 
clearly in the figure to permit the instrument 
to be built, further comment is not required, 
except that the meter M has 100-0-100 pa 
movement. 

An indicator with simpler circuit elements 
has been used successfully at NRL. The circuit 
diagram for their instrument is reproduced in 















SPECIFICATIONS FOR SINGLE CRYSTALS 


297 


Figure 32. The circuit is novel in that the low B 
potential is placed on the control grids. The 
voltage generated by pressing on the crystal is 
placed upon the two screen grids. Better sta¬ 
bility than that obtained from the usual tube 


from making whatever tests may be required 
to establish acceptability. Hence, the selection 
of crystals for transducers from the standpoint 
of discarding any substandard ones will be dis¬ 
cussed at some length. 



Figure 31. Circuit diagram of the UCDWR polarity indicator. 


connections is claimed. Since the writer has 
neither built nor had experience with this equip¬ 
ment, no critical comment can be added. 

SPECIFICATIONS FOR SINGLE 
CRYSTALS 

Official specifications for piezoelectric crys¬ 
tals are not in a satisfactory state at the present 
time. In fact, as far as the writer is aware, 
such specifications do not even exist in the case 
of RS. In the discussion of various properties 
amenable to quantitative specification in the 
sections to follow, an attempt will be made to 
point out reasonable expectations for the sev¬ 
eral characteristics and to indicate the nature 
of inspection tests to be performed. 

The UCDWR Laboratory has purchased all 
of its crystals from outside sources. However, 
this dependence on a commercial source of 
crystalline material, even though the quality 
of the piezoelectric crystals so obtained may be 
consistently good, does not free a laboratory 


' Visible Defects 

Visible defects in crystals are of two sorts, 
veils and chips. Veils occur within the body of 
crystals and are caused by unfavorable satura¬ 
tion conditions in the solution during the 



Figure 32. Circuit diagram for NRL polarity 
indicator. 

growth of the mother bar. Improper saturation 
conditions may be caused by either an incorrect 
temperature or inadequate circulation. These 
veils are of two distinct types, the one being 
associated with a supersaturation condition and 
the other with an unsaturated condition. In 
cutting 45 degree Z-cut ADP plates from the 


































































298 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


mother bar, it is especially important to exclude 
the seed-cap region with its usually prevalent 
veils. 

It is not uncommon for veils to contain small 
quantities of the original saturated salt solu¬ 
tion. Their presence leads to voltage breakdown 
and crystals containing them must be discarded. 
According to the Navy specification for ADP-’ 
all crystals having appreciable veils as gauged 
by eye shall be rejected unless they pass suc¬ 
cessfully a voltage-breakdown test. This test is 
detailed in Section 8.5.8. Where crystals are to 
be used in transducers at operating voltages in 
excess of those specified in the Navy voltage 
test, it is clear that they should be tested at the 
higher operating voltage in question. 

With respect to crystals which possess minor 
flaws, such as chipped corners or slight surface 
imperfections, the question as to whether they 
shall be used is more difficult to decide. Accord¬ 
ing to the Navy specification'-* mentioned, fin¬ 
ished crystal plates with minor flaws and im¬ 
perfections shall be acceptable provided they 
meet certain standards of appearance and also 
meet the electrical and mechanical performance 
requirements enumerated in a later section. The 
Navy appearance standard consists of photo¬ 
graphic views of chipped crystals, the chips 
being of various sizes. Crystals with very small 
chips were considered acceptable, those with 
larger chips were labeled “no go.” The size of 
the chips for acceptable crystals seemed to be 
about Yig or %2 in. in diameter and consider¬ 
ably less in depth. The crystals in the category 
labeled “no go” had larger and deeper chips, 
or perhaps a whole corner would be chipped off 
across the entire thickness dimension. 

Much difference of opinion exists on the ques¬ 
tion as to when a crystal is or is not acceptable 
owing to the presence of chipped regions. From 
the standpoint of their performance in a trans¬ 
ducer, the existence of chipped corners may not 
be serious. According to the observations of 
W. P. Mason, communicated in conversation, 
the location of voltage breakdown points in de¬ 
fective transducers were not correlated with 
the presence of chips or minor surface flaws in 
individual crystals. Inasmuch as a high per¬ 
centage of the crystals obtained from a mother 
bar suffer from visible flaws incurred during 


processing, greater attention should be given 
to setting up definitive standards based on 
actual performance data. 

In an experimental laboratory, where crystals 
are constantly being cut to particular size speci¬ 
fications, any relatively large crystal containing 
chipped edges may be salvaged and recut for 
use in another transducer employing a higher 
resonant frequency. 


" (xeometric Tolerances 

In the case of ADP crystals, certain standard 
dimensions were designated during World War 
II by BuShips. With respect to length, the 
crystals were either 1.10 or 1.25 in.; with re¬ 
spect to width, either V 2 or 1 in.; and with 
respect to thickness, either Yu, l^, or 1/0 in* 
Tolerances in either length, width, or thickness 
were specified as ±0.005 in. This tolerance 
specification is regarded as a liberal one. A 
tolerance as low' as ±0.001 in. still would be 
considered reasonable although it might involve 
a slight extra charge. 

Angular tolerances were not explicitly stated 
in the specifications'* set up by BuShips. How¬ 
ever, they were implied when limiting values 
were placed upon electrical characteristics, a 
discussion of which is contained in sections to 
follow. Crystal tolerances with regard to square¬ 
ness of cut might reasonably be held to ±1 de¬ 
gree, but the tolerance of the orientation angle 
cannot readily be held to such a low value. How¬ 
ever, a tolerance in orientation of ±2 degrees is 
a reasonable one. 

Angular orientation errors do not offer any 
particular difficulty in the case of ADP crystals 
since the rate of change of the dielectric con¬ 
stant with respect to angle is a slowly varying 
function. In the case of X-cut and Y-cut RS, 
however, there is a very large difference in the 
dielectric constant between the X and Y axes. 
Particularly in Y-cut crystals, slight deviations 
from the correct angle of orientation can pro¬ 
duce a marked difference in the capacity of the 
crystal and a routine inspection of all Y-cut 
crystals may be necessary. The best test is 
probably a capacity check as discussed in Sec¬ 
tion 8.5.4. 





SPECIFICATIONS FOR SINGLE CRYSTALS 


299 


8.5.3 Poiaj.j 2 ;ecl Light and X-Ray Diffraction 

The fact that piezoelectric crystals are bire- 
fringent has led individuals to make use of 
polarized light in an effort to determine axes 
of orientation. The optic axes of both RS and 
ADP lie in the longitudinal dimension of the 
mother bars. In the case of 45-degree Z-cut ADP 
crystals the electric field is applied in the same 
direction as the optic axis. 

The simplest type of polarized light observa¬ 
tion may be made with some such equipment 



Figure 33. Instrument for observing crystals 
in plain parallel polarized light. 


as is illustrated in Figure 33. This piece of 
apparatus consists of a polarizer and an ana¬ 
lyzer, each made up of a large sheet of Polaroid. 
It is customary to rotate the analyzer until the 
field is dark. If a 45-degree Z-cut ADP crystal 
is laid with its electrode face on the polarizer, 
an observer would be looking in the direction 
of its optic axis. Consequently, the field would 
remain dark even though the crptal were ro¬ 
tated 360 degrees about its optic axis. If the 


crystal were laid on its long edge or on its end, 
an observer would see alternating periods of 
light and darkness during a 360-degree rota¬ 
tion. Hence, this simple observation with rudi¬ 
mentary equipment could reveal whether crys¬ 
tals were cut an entire quadrant off from the 
correct orientation. 

In the case of 45-degree X-cut and 45-degree 
Y-cut RS crystals, the orientation does not per¬ 
mit observation along the optic axes when the 
crystals are simply laid on the polarizing plate. 
Therefore, alternating periods of darkness and 
light will be observed on the analyzer when ro¬ 
tation occurs about any of the three axes of 
either type of crystal. Consequently, it would 
be necessary to resort to more complicated ar¬ 
rangements to secure any precise data on the 
orientation. In view of the fact that all these 
crystals may readily be cut to an accuracy of 
2 or 3 degrees, it would not seem particularly 
useful to proceed to more refined polarized light 



Figure 34. Etch figures on a Rochelle salt sur¬ 
face normal to the Z axis. In each case, the X 
axis is horizontal, the Y axis is vertical. 

methods of observation. On the other hand, 
observation of RS crystals in convergent polar¬ 
ized light might be desirable in that unsym- 
metrical patterns would appear if the angle of 
cut were incorrect by a small amount. 

For an accurate check on the correctness of 
angular cut for any of the crystals under dis¬ 
cussion, access should be had to X-ray diffrac¬ 
tion equipment. While the X-ray examination 
of each crystal which enters into the construc¬ 
tion of a transducer is normally not justified, 
occasion may arise when it would be desirable 



















300 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


to test sample crystals by this method. The NRL 
has installed X-ray diffraction equipment for 
this purpose but such facilities have not been 
available at UCDWR. 

The following account of etching tests made 
by Cadyi® may be helpful in distinguishing the 
axial directions in RS plates. Very character¬ 
istic figures are easily produced by lightly mois¬ 
tening a polished crystal surface. After drying, 
a face normal to the X axis is found to be 
covered with fine striations parallel to the Z 
axis. On faces normal to the Z axis, minute 
rectangular pyramids (“etch hills”), sometimes 
truncated, extend upward from the surface. 
Some characteristic forms for the X-Y plane 
are shown in Figure 34 as viewed from above. 
The X axis bisects the projection on the X-Y 


in the crystal. Any of several well-known 
methods may be employed for accurately meas¬ 
uring these small values of capacitance to the 
desirable tolerance of ±0.1 [.qif in the capaci¬ 
tance range of 5 to 30 [.qif. 

Inspection of the equivalent electric circuit of 
a crystal, shown in Figure 37, indicates that 
the proper place to measure Co might be at a 
point considerably above the resonant fre¬ 
quency of the crystal, since at that point the 
impedance of the series LCR branch would be 
so high that the impedance of the whole crystal 
unit would be effectively that of Co alone. 
Difficulties arise, however, owing to the fact 
that a crystal is not singly resonant over a wide 
range of frequencies, e.g., at certain frequen¬ 
cies, modes of vibration other than the longi- 


Table 1. Navy specifications^ for some electrical characteristics for ADP crystals. 


Difference between 


Dimensions 

(in.) 

Resonant 

frequency 

(kc) 

Capacity at 

1,000 cycles 
(/^Mf) 

Capacity 

ratio 

antiresonant and 
resonant 
frequencies (kc) 

1.10 X 1.00 X 0.250 

48.4 ± 0.5 

15.7 ± 1.0 

15.2 ± 1 

1.57 ± 0.1 

1.10 X 1.00 X 0.500 

48.4 ± 0.5 

7.8 ± 0.5 

15.0 ± 1 

1.58 ± 0.1 

1.25 X 0.5 X 0.125 

50.3 ± 0.5 

17.8 ± 1.0 

14.1 ± 1 

1.75 ± 0.1 

1.25 X 0.5 X 0.250 

50.2 ± 0.5 

8.9 ± 0.5 

14.1 ± 1 

1.75 ± 0.1 

1.25 X 0.5 X 0.500 

50.0 ± 0.5 

4.5 ± 0.5 

14.3 ± 1 

1.72 ± 0.1 


plane of the acute angle alpha, which has a 
value of roughly 60 degrees. On a face normal 
to the Y axis, the pyramids are of the same 
general nature as illustrated in Figure 34, the 
longer dimensions of the base being in most 
cases parallel to the Z axis. Owing to the strong 
polarity in the X direction, one might expect 
marked differences in the etch figures on oppo¬ 
site sides of an X-cut plate. On the contrary, the 
striations look just alike. 


Capacitance 

An important electrical property of a piezo¬ 
electric crystal is its static capacitance Co, 
which represents the capacitor formed by the 
dielectric of crystalline material between the 
two electrodes. Some of the factors which affect 
the measured value of Co are: angular orienta¬ 
tion, linear dimensions, temperature, crystal 
holder, and irregularities such as chips or flaws 


tudinal mode will be excited as well as higher- 
order harmonics of any of the possible modes. 
Therefore measurements of Co obtained near 
any of the higher response frequencies would 
be in error. This difficulty of avoiding higher- 
order resonances at frequencies above the fun¬ 
damental longitudinal resonance, has led to the 
practice of measuring the capacitance at low 
frequencies, usually at 1,000 c. Since the react¬ 
ance L is negligible at this low frequency, in¬ 
stead of Co, the capacitance measured will be 
Cy, where Cy is equal to Co + C. 

For 45-degree Y-cut RS and 45-degree Z-cut 
ADP, Cy has a value about 9 per cent higher 
than Co alone. The Navy specifications® for ADP 
include the value of Cy at 1,000 c and allow a 
tolerance of from 5 to 10 per cent depending 
upon the size of the crystal. Table 1 gives the 
Navy specifications for the capacitance at 
1,000 c for five common sizes of ADP crystals. 
In addition, it also gives values for the resonant 
frequency, the capacity ratio, and for the dif- 


ESTRIC1 











SPECIFICATIONS FOR SINGLE CRYSTALS 


301 


ference between the antiresonant and resonant 
frequencies. The meaning of these latter quan¬ 
tities will be discussed further in subsequent 
sections. It has been found by McSkimin^- that 
reasonable values for the capacitance of ADP 
crystals at 1,000 c are given by the following 
equation: 

^ 1.38/-U? 

L r = -^- mmi (dimensions in cm) (1) 

where I, w, and t, represent the length, width 
and thickness of the crystal, respectively, in 
centimeters. 

For 45-degree Y-cut RS crystals, the capaci¬ 
tance may be calculated to a fair approximation 



Figure 35. Capacitance bridge for measuring 
Co. 


by substituting the constant 0.87 for the 1.38 
of equation (1). Since the effective dielectric 
constant of Y-cut RS varies markedly with 
angular orientation, the capacitance measure¬ 
ment serves as a valuable check on the correct¬ 
ness of the cut. Some published tabular data^^ 
on the capacitance of about 600 commercial 
Y-cut crystals indicate that fluctuations of as 
much as ±7 per cent from the average value 
given by the above equation may be expected. 

For 45-degree X-cut RS, such simple calcula¬ 
tions of capacitance are impossible. The dielec¬ 
tric constant varies markedly, not only with 
temperature but also with the applied field 
strength, and in neither case in a monotonic 
fashion. Reference is made to a paper by Fro- 
man^^ for graphical data on X-cut RS. 

A simple bridge circuit readily adapted to 
this type of testing appears in Figure 35. Ca¬ 
pacitance measurements of sufficient accuracy 
are obtained by incorporating a substitution 


method into the bridge. The resistance arms 
Ri and R 2 are equal and C can be any high 
quality capacitor (variable or fixed) capable of 
balancing the other bridge arm containing a 
calibrated standard C^. The high-impedance 
voltmeter V should have sufficient sensitivity 
to give sharp indication of balance. In operation 
the crystal, supported in a suitable holder, is 
shunted across C^. A balance is obtained by 
varying either or C. Then the crystal is re¬ 
moved and the bridge rebalanced by varying C^. 
The value of Cj, is the difference between the 
two settings of C,. 

Another satisfactory method of determining 
Cy is to employ the admittance measuring cir¬ 
cuit shown in Figure 38 of Section 8.5.5. In this 



Figure 36. Special crystal holder for reducing 
stray capacitance. (Bell Telephone Labora¬ 
tories.) 

circuit the magnitude of crystal admittance may 
be computed, and hence Cr^, since the voltage 
across the crystal, the current through the crys¬ 
tal, and the applied frequency are known. Ob¬ 
servations made between 1 kc and one-half the 
resonant frequency /,. will show that within the 










































302 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


limits of measurement the crystal behaves as a 
capacitor having an admittance of to (Co + C). 

Where a high degree of accuracy is necessary, 
reliable shielding and grounding techniques 
must be used throughout the test circuit and 
for the crystal holder. Not only errors in abso¬ 
lute values but also relative errors owing to 
temperature variations may be minimized by 
reducing stray capacitance and grounding ef¬ 
fects of the crystal holder, and by careful choice 
of the materials used in constructing the holder. 
The effect of the stray capacitance is to cause 
an apparent increase in Cj.. The effect of ground 
proximity is to disturb the fringing flux dis¬ 
tribution around the crystal and therefore to 
lower the measured value of Cj,. In general, 
errors due to ground proximity are greater 
for crystals of larger thicknesses and for the 
smaller widths. A special crystal holder de¬ 
signed by McSkimin^- is illustrated in Figure 
36. The grounding effect is controlled and stray 
capacitance is reduced to about 0.01 p^if by 
making the electrical connections through 
shielded cables to a carefully isolated crystal. 


Admittance and Q 

An inspection of the simple equivalent circuit 
for a piezoelectric crystal, shown in Figure 37, 
indicates the existence of a critical frequency 



Figure 37. Representation of a crystal and its 
equivalent circuit. 


at which the mechanical LCR arm of the 
network will exhibit a maximum admittance. 
At frequencies increasingly higher than the 
admittance rapidly decreases until a minimum 
value is reached at the frequency f„. The abso¬ 
lute value of the admittance at /„ is usually so 
small and the minimum so flat for the funda¬ 
mental longitudinal response of a crystal that 


a precise determination of /„ is difficult. As the 
frequency increases above /„, the admittance 
increases until it approaches the value of coCo, 
i.e., the crystal appears as a capacitor with the 
capacitance Co. It was shown in Section 8.5.4 
that at frequencies considerably below the 
crystal appeared as a capacitor with the capaci¬ 
tance Cf (= Co C). 

The salient features of an admittance-versus- 
frequency curve are shown in Figure 39 for an 
ADP crystal with the dimensions lV 2 xlxl^ in. 
The first maximum occurring at 39.2 kc repre¬ 
sents resonance for the fundamental longitu¬ 
dinal vibration of the crystal, its accompanying 
antiresonance appearing at 40.6 kc. A number 



Figure 38. Admittance circuit: two voltmeter 
method. 


of maxima and minima corresponding to higher- 
order resonances and antiresonances are also 
shown. At frequencies well below the admit¬ 
tance is primarily a function of the capacitative 
reactance and therefore increases 6 db per oc¬ 
tave. The dashed line corresponds to the ad¬ 
mittance for the true static capacitance Co- An 
experimental approximation to the mechanical 
Q of a crystal Q„, is obtained by dividing by 
the difference in the frequency settings required 
in order to reduce the admittance 3 db on either 
side of /,„. 

The circuit employed at UCDWR for admit¬ 
tance measurements is reproduced in Figure 38. 
A constant voltage Vi is applied to the crystal 
and then, as the frequency of the generator is 
varied, the current through the crystal is ob¬ 
served in terms of the voltage drop Vo devel¬ 
oped across a relatively small resistor Ri in 
series with the crystal. Some error will be intro¬ 
duced by using the same value of Ri through¬ 
out an admittance test but for the sake of the 
general picture this may usually be neglected. 
A more detailed discussion of the absolute ad- 















SPECIFICATIONS FOR SINGLE CRYSTALS 


303 


mittance measurement is given in Section 9.1.1, 
but it will be noted here that the variable- 
frequency generator must meet several definite 
requirements. Briefly, it should operate from 
1 to 150 kc with a stability of ±20 c at all fre¬ 
quencies, be capable of adjustment in incre¬ 
ments of 1 or 2 c, have a low output impedance, 
and have an extremely low harmonic content. 
The construction of the crystal holder must be 
such as always to enable the crystal to be sup- 


Typical values of for actual transducers 
range from perhaps 10 down to 2. 

Resonant Frequencies 

The two resonant frequencies of major inter¬ 
est may be termed “series resonance” and 
“antiresonance” /„. Series resonance or simply 
“resonance” is defined as the frequency at which 
the series LCR branch of the equivalent circuit 



Figure 39. Typical admittance curve for single crystal. 


ported in exactly the same manner if repro¬ 
ducible measurements are to be obtained. 

No specifications have been set up for ADP 
or RS crystals in terms of their admittance 
characteristics. Since the values are obtained 
from air measurements they probably have little 
bearing on the behavior of loaded crystals op¬ 
erating under water. In the latter case, for ex¬ 
ample, is quite small compared to the high 
values characteristic of crystals freely vibrat¬ 
ing in air. In reported measurements^^ on a 
total of several hundred 45-degree Y-cut RS 
crystals, divided into ten groups on the basis 
of their dimensions, average values for Q,„ of 
the various groups ranged from 3,990 to 6,950. 


(see Figure 37) appears as a pure resistance, 
i.e., the net series reactance is equal to zero. 
Antiresonance occurs at a slightly higher fre¬ 
quency when the series LCR branch exhibits a 
net inductive reactance and is defined as the 
frequency at which the susceptances are equal 
and opposite in the two parallel branches of the 
equivalent circuit. 

The frequencies /,. and are related to the 
frequencies /„, and /,^ corresponding to maxi¬ 
mum and minimum admittance as follows: 

f, - f. = and ^ (2Co - C). 

In the practical case where crystals are meas- 

































304 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


ured in air, the terms in R- are negligible so 
that and /„ = /„. Thus, within the limits 

of experimental measurement, series resonance 
is indistinguishable from the point of maximum 
admittance and antiresonance is indistinguish¬ 
able from the minimum admittance frequency. 

In the Navy specifications^ for certain stand¬ 
ard sizes of ADP crystal plates, the following 
statement occurs: 

The oscillatory characteristics of the various crystal 
plates shall be determined by means of their respective 
resonant and antiresonant frequencies. The values of 
resonant frequency, capacity at 1000 cycles per second, 
and capacity ratio shall be within the limits given in 
Table 1. The values of capacity at 1000 cycles are for 
direct crystal capacity and are exclusive of capacities 
to ground of either electrode. The values of capacity 
ratio are determined from the resonant and anti¬ 
resonant frequency measurements by the formula: 

..Co 1 

Capacity ratio =-q = ‘ 

It will be observed that Table 1 lists tolerances 
for /,. of ±0.5 kc and for the ratio of capaci¬ 
ties, Co/C, of ±1. As an alternative specifica¬ 
tion to the capacity ratio given in Table 1, the 
crystals may meet a specification for the differ¬ 
ence between the antiresonant and resonant 
frequencies with a tolerance limit of ±0.1 kc. 

Although no specifications on RS crystal 
plates with regard to resonant frequency are 
known to the writer, available measurements^^ 
on several hundred 45-degree Y-cut plates give 
an indication of the tolerances to be expected 
in commercial lots. It appears from these data 
that a tolerance of ±1 per cent on resonant 
frequency would be quite reasonable, especially 
for crystals resonating at 55 kc or less. In fact, 
for the latter group, ±0.5 per cent could be met. 
In this same report,values of R at resonance 
are also stated for 45-degree Y-cut RS. In gen¬ 
eral there seems to be no point in measuring R, 
unless perhaps in the case of spliced crystals. 

With slight modifications the circuit used for 
measuring admittance (Figure 38) may be used 
for determining /,., /„, and R. In the circuit of 
Figure 40, Ri and Ro have values between 
10 and 100 ohms. The detector is a sensitive, 
high-impedance device, and the variable fre¬ 
quency oscillator meets the requirements out¬ 
lined in Section 8.5.5 for stability, etc. 


In the measurement of /,. and /„, a constant- 
amplitude low voltage is applied across Ri 
from the oscillator. Then as the frequency of 
the driving voltage is varied, the frequencies 
corresponding to the points of maximum and 
minimum current through the crystal are ob¬ 
served. Unless absolute values of voltage or 



Figure 40. Useful circuit for determining /r, fa , 
and R. 


admittance are required an uncalibrated detec¬ 
tor is all that is needed in measuring /,. and /„. 
These two important frequencies should be 
measured to as high an accuracy as the fre¬ 
quency stability of the oscillator and the sensi¬ 
tivity of the detector will allow, preferably to 
one part in fifty thousand or better. 

The resistance of the series LCR branch of 
the crystal equivalent circuit may be readily 
measured also with the test circuit of Figure 40 
by the substitution method. The procedure is as 
follows: With the test crystal in position, the 
frequency of the driving oscillator is carefully 
adjusted to the series resonant frequency of the 
crystal and the exact reading of the indicator 
on the detector noted. The crystal is now re¬ 
moved from its holder and a variable resistor 
substituted in its place. While holding the oscil¬ 
lator frequency at /^, adjust the value of the 
variable resistor until the reading of the detec¬ 
tor is identically the same as before. Since the 
impedance of the crystal reduces to R at the 
value of R is equal to the value of the substi¬ 
tuted resistor. Either a calibrated detector or 
a vacuum-tube voltmeter is desirable, but not 
necessary, in measuring R. Since the substitute 
resistor should be noninductive, a more prac¬ 
tical solution may be to use a series of accu¬ 
rately known fixed resistors of which nonin¬ 
ductive types are available. The substitution 
method of determining R permits greater accu¬ 
racy in practical test circuits than a maximum- 
and minimum-admittance method. 









CEMENTS 


305 


A direct instrumental method for measuring 
the capacitance ratio Co/C has been developed 
at the BTL and is discussed in their report on 
ADP.' 

® ‘ D-C Resistance 

According to Navy specifications*^ for ADP 
crystal plates, “the D.C. volume restivity be¬ 
tween the electroded faces of the crystal plate 
shall be not less than the following for the 
grades specified”: [quotation includes follow¬ 
ing table] 


Resistivity at 25 C 

Grade ohm-centimeter 

AAA 1.0 X 1010 

AA 0.9 X 100 

A 2.5 X 108 


Since the volume resistivity of ADP depends 
primarily on the purity of the supersaturated 
solution in which the crystals are grown, it is 
necessary to test only a few representative sam¬ 
ples from each batch and not to make resistance 
tests on every finished crystal plate. 

A more detailed discussion of both RS and 
ADP with respect to electric resistivity occurs 
in Sections 8.2.4 and 8.2.8, respectively. 

« " « High Voltage 

According to Navy specifications** for ADP 
plates, “Crystals of grades AA and AAA shall 
be capable of withstanding voltage gradients of 
20,000 volts per inch of thickness at a frequency 
approximately one-half the resonant frequency 
given in Table I [Table 1 of this chapter] for 
the size of crystal being tested. For this test, 
the crystal should be submerged in a suitable 
fluid (carbon tetrachloride is one such fluid), 
and the voltage shall be maintained for a period 
of at least 30 seconds.” The plates will be con¬ 
sidered as having met this requirement if a 
suitable sampling does so. 

Since clean ADP crystals i/4 in. thick will 
usually withstand 20,000 v rms at 60 c, the 
above specification is not a stringent one. Ro¬ 
chelle salt crystals would also readily pass this 
specification. Should a particular application 


require crystals to be driven at a voltage higher 
than the Navy specifications given above, then 
each crystal should be tested well in excess of 
the actual operating voltage. 

Although no design of high-voltage test equip¬ 
ment is given here, reference is made to special 
equipment designed for this purpose at BTL.‘ 
Its main feature was a resonant high-voltage 
circuit involving the capacitance of the crystal 
under test; in case of crystal breakdown, the 
circuit characteristic changed in such a manner 
as to reduce the applied voltage. 

CEMENTS 

The quality of the cement joint with which 
piezoelectric crystals are attached to supporting 
structures is one of the most important consid¬ 
erations in the construction of transducers, yet 
there is practically no agreement as to the best 
technique. Much of the discussion on cementing 
procedures must be written in the subjunctive 
mood. Even the choice of a cement cannot be 
made on a conclusive basis and some workers 
in the field go so far as to say that almost any 
cement will be satisfactory if the proper tech¬ 
nique for its application is once developed. 

Extreme pessimism on the subject of cements 
is probably not justified, although at best it is 
admittedly difficult to standardize techniques. 
In the absence of unanimity among the various 
laboratories engaged in this type of work, it 
seems best to describe in some detail some of 
the more customary methods in use at the pres¬ 
ent time. In addition, the special requirements 
peculiar to this problem will be outlined and 
discussed. 

Specifications 

Several of the specific requirements which 
must be met by any cement intended for attach¬ 
ing crystals to supporting structures will be 
enumerated and briefly discussed. 

1. The cement must not interact with the 
crystalline material. This requirement precludes 
the use of a water soluble cement for either 
ADP or RS. With RS it is also necessary to 
avoid dehydration. For the most part, the nu¬ 
merous plastic cements which require an accel- 












306 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


erator are likewise excluded from consideration, 
since practically all of them contain either an 
acid or an alkaline catalyst. Preliminary trials 
at UCDWR with several plastic cements con¬ 
taining catalysts were unsuccessful. 

2. The cement must have a low electrical 
conductance. 

3. The cement must be capable of being cured 
within certain critical temperature ranges. In 
the case of RS the upper temperature limit is 
50 C, so that most thermosetting and thermo¬ 
plastic cements cannot be used. With ADP crys¬ 
tals the upper temperature limit is in the neigh¬ 
borhood of 135 to 140 C. This limit is not par¬ 
ticularly restrictive and does permit the use of 
both thermoplastic and thermosetting cements. 

4. The cement must produce a joint in which 
only a small loss of mechanical energy can occur 
while the crystal is being driven. This energy 
loss should not increase greatly when high 
strains exist in the cement layer. This require¬ 
ment is very important but fortunately it is 
subject to scientific measurement.^^ 

5. The cement must be capable of taking care 
of the differential coefficient of expansion ex¬ 
isting between the crystals and their supporting 
structure. The severity of this requirement de¬ 
pends on the actual specifications which are 
imposed for a given application. As the tend¬ 
ency is to set increasingly wider limits upon 
temperature performance, especially by extend¬ 
ing the low temperature end, it has become more 
difficult to find an adequate cement. 

Although there are literally thousands of ce¬ 
ment compositions available commercially, most 
of these consist of modifications of perhaps a 
dozen basic types. It is highly probable that the 
best cement composition for crystal applications 
has not yet been developed. Furthermore, it is 
clear that not one, but several cements will be 
found exceedingly useful for the various com¬ 
binations of crystalline materials and support¬ 
ing structures encountered. 


Application Conditions 

Following the selection of a cement which 
possesses the best combination of desirable 
properties for a given application, it will be 


discovered that only the first of a long series 
of difficulties has been met. The quality of a 
cement joint has been found to depend on nu¬ 
merous variables. 1'’ Several of the factors in¬ 
volved will be discussed in this section, others 
in Section 8.6.3. 

The Method of Applywg the Cement. The 
possibilities that exist are spraying, brushing, 
dipping, and roller coating. A choice between 
these methods may depend in part on the 
amount of work to be done, but more important 
is the quality of the joint produced and, in the 
interest of uniformity, the ease of duplicating 
performance from day to day. It has been found 
in practice that some cements are best adapted 
to a given method of application; for others, 
the method may be immaterial. 

The Amount of Cement. Quantity is a critical 
factor. Generally speaking, the cement layer 
should be thin, but if too thin there may not 
be sufficient accommodation for differential 
thermal expansion and cracking of the crystal 
results. If too thick, the loss of mechanical en¬ 
ergy in the cement layer becomes excessive. 
Standardization of cement layers to a given 
thickness is most difficult because it depends 
on the method of application, on the judgment 
and operating skill of the technician, on the 
temperature and pressure and duration of the 
curing process, and on the maintenance of a 
given consistency in the original cement. These 
individual factors will be discussed later in 
some detail. 

The Condition of the Cemented Surfaces. 
With regard to the condition of the crystal 
surface, two factors need be mentioned. With 
some types of cement it is entirely satisfactory 
to have a smooth surface on the crystal. This 
is particularly true for thermoplastic and ther¬ 
mosetting cements in which the solvent is per¬ 
mitted to escape before the surfaces are joined. 
Although there is a question of individual pref¬ 
erence between smooth versus sanded surfaces, 
it would seem that a roughened surface should 
give a better bond as a general rule. The surface 
of the supporting structure will vary with the 
type of material. Metallic surfaces, in general, 
will be smooth. Where insulating wafers are 
used between crystals and backing plates it is 
desirable to provide a path for the solvent in 



CEMENTS 


307 


the cement to escape. This can result from the 
use of either porous wafers or insulating wafers 
containing specialized channels for this pur¬ 
pose. It is obvious that surfaces should be thor¬ 
oughly cleaned before the application of an 
adhesive, either by a light sanding operation as 
shown in Figure 17 or by wiping with a suit¬ 
able organic solvent. 

Humidity. Humidity is a factor, especially in 
the case of RS. A humidity-temperature curve 
which gives the upper and lower limits of sta¬ 
bility for RS is reproduced in Figure 1. It is 
essential to work slightly below the lower limit 
curve shown on this graph so that the crystal 
surface will not adsorb any appreciable mois¬ 
ture at any time during the cementing and cur¬ 
ing process. Unless all surface moisture is re¬ 
moved from RS crystals before cement is ap¬ 
plied, the resulting bond will be very poor. It 
is also important to avoid dehydration of the 
salt. This means humidity control in accordance 
with the data recorded on the humidity-temper¬ 
ature graph. 


Curing 

Some type of controlled curing oven is essen¬ 
tial. With ADP, only temperature need be regu¬ 
lated and the highest temperature needed is 
not in excess of 150 C. With RS, it is necessary 
to control the relative humidity as well. Since 
commercial equipment entirely suited to both 
temperature and humidity control is available, 
detailed information on suitable laboratory ap¬ 
paratus of this kind is omitted from the present 
discussion. 

With regard to the conditions which must 
exist during the curing process, the discussion 
may be confined to two main headings. 

1. The pressure on the cemented surface must 
be controlled in order that a proper thickness 
of cement layer will result. Pressures as high 
as 200 psi may be in order, especially for large 
arrays. The actual pressures applied in a par¬ 
ticular case must be correlated with tempera¬ 
ture. At high temperatures, too high a pressure 
will result in most of the cement being squeezed 
out. Thus it will be seen that a fairly critical 
control of the pressure is demanded. Additional 


remarks on this topic will accompany the de¬ 
tailed directions for the use of individual types 
of cement. 

2. The temperature and duration of the cur¬ 
ing process must be adapted to the particular 
type of cement being employed. With some 
types of cement, curing consists essentially of 
a drying process during which the solvent 
escapes. The rate of escape of the solvent de¬ 
pends on temperature, time, the area involved, 
the nature of the solvent and the porosity of 
the surfaces. It is usually recommended that 
an appreciable amount of the solvent be evap¬ 
orated before the two surfaces come together. 
This results in a tacky condition of the surface 
and hastens the subsequent drying process. The 
most satisfactory temperature-time relationship 
for securing a satisfactory tacky condition can 
be determined experimentally by trying differ¬ 
ent combinations and then testing the quality 
of the bonds obtained. 

In thermoplastic or thermosetting bonds it is 
customary to permit all the solvent to escape 
before the surfaces are in contact. In this case 
the application of heating either results in a 
sufficient softening of a thermoplastic cement 
or in a proper degree of polymerization of the 
thermosetting cement. Data covering the proper 
temperature and pressure in these cases can 
usually be obtained from detailed instructions 
furnished by the manufacturer. Accordingly, 
they are much more amenable to control. 


8.6.1 Viilcalock and Bakelite Cements 

In the construction of transducers the two 
adhesives most commonly used have been Vulca- 
lock and bakelite cement. At UCDWR the bake¬ 
lite cement used had the number BC-6052. In 
an early report^^ from BTL bakelite cement 
BC-8723 was recommended. Vulcalock and 
bakelite BC-6052 cement seem to be very simi¬ 
lar with possibly some difference in the com¬ 
position of the solvent. Both seem to be based 
on a natural-rubber component and both yield 
sufficiently to prevent the cracking of crystals 
when subjected to low temperatures, except in 
extreme cases. 

In the application of either of these cements 



308 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 



Figure 43. Figure 44. 

Contrasting- effects of —40 C temperature of Vulcalock (Figures 41, 42) and Butyl-C (Figures 43, 44) 
cement joints between ADP crystals and insulating wafers of either Durez plastic (Figures 41, 43) or 
porous ceramic (Figures 42, 44). The other side of each wafer was bonded to a steel plate with XCU 
16257 urea formaldehyde cement. Following their preparation, all specimens were subjected to a vacuum 
and aged in castor oil for seven days at 60 C. Cooling to —40 C took place in either 3 hours from 30 C 
(Figures 41, 43) or 3 hours 20 minutes from 28 C (Figures 42, 44). With Vulcalock bonds, first cracks 
appeared at either 10 C (Figure 41) or 13 C (Figure 42) ; with Butyl-C bond, a tiny crack appeared in 
one crystal at —39 C (see arrow in Figure 44), the other remained intact (Figure 43). (Bell Telephone 
Laboratories.) 




















CEMENTS 


309 


the best method^^ seems to be to apply a thin 
uniform layer of cement on both surfaces and 
allow it to air dry for a definite length of time. 
The cement should always possess the same con¬ 
sistency initially and drying should take place 
at a prescribed temperature and for a definite 
length of time, since the thickness or tackiness 
of the cement is a controllable function of the 
drying time. A cement of satisfactory consist¬ 
ency, at the time of pressing the specimens to¬ 
gether, has a composition of about 60 per cent of 
solid matter by weight. A pressure as high as 
200 psi can be maintained for a few minutes 
without forcing out too much of the cement and 
results in a good bond. The pressure should then 
be reduced to 30 psi and the specimens placed in 
a drying oven. This reduction of pressure is nec¬ 
essary to prevent further loss of cement when 
the increased temperature of the oven causes it 
to become more fluid. However, it is definitely 
beneficial to compress the cement as it contracts 
from loss of solvent. Therefore, the pressure 
should not be reduced too much. The length of 
time for curing either bakelite or Vulcalock ce¬ 
ment should be at least 24 hr and even longer 
times are beneficial. The temperature in the 
case of RS may be 40 C with the relative humid 
ity 50 per cent. For ADP the temperature may 
be higher, even 80 C, and the drying time need 
not be as long. 

The unsatisfactory use of Vulcalock for ce¬ 
menting ADP crystals to supporting structures 
in transducers which must operate at extremely 
low temperatures is photographically depicted 
in Figures 41 and 42. In Figure 41 an ADP 
crystal has been bonded with Vulcalock cement 
to a Durez wafer and in Figure 42 to a porous- 
ceramic wafer. In both cases the wafers were, 
in turn, bonded to steel plates with the catalyzed 
urea formaldehyde cement XCU 16257. The use 
of the latter cement is discussed in some detail 
in Section 8.6.8. The ADP crystals in these two 
illustrations were treated in essentially the 
same manner, having been subjected first to a 
vacuum and then aged in castor oil at 60 C for 
7 days. At the end of this period the crystals 
were cooled from around 30 C down to —40 C in 
approximately 3 hr. In Figure 41 the first cracks 
in the crystal appeared at 10 C and in Figure 42 
at 13 C. This behavior of crystals bonded with 


Vulcalock cement may be contrasted with that 
of crystals bonded with Butyl-C cement, dis¬ 
cussed at length in Section 8.6.5, by a direct 
comparison of Figure 41 with Figure 43 and of 
Figure 42 with Figure 44. 

In connection with the use of Vulcalock ce¬ 
ment it will be of interest also to read the com¬ 
ments in Section 8.2.4 concerning the trapping 
of moisture beneath cement layers. The effect 
on the leakage resistance of RS crystals coated 
with Vulcalock cement is brought out graph¬ 
ically in Figure 3. This graph emphasizes the 
necessity for the complete removal of all ad¬ 
sorbed water before the application of the ce¬ 
ment. This is probably best accomplished by 
subjecting the crystals to a vacuum for a few 
minutes just previous to the application of the 
cement. 

The question of the compatibility between the 
cement and the transducer liquid in which crys¬ 
tals will be immersed must also be considered. 
Both Vulcalock and bakelite cements may be 
used with DB grade castor oil. In other liquids, 
for example, mixtures of castor oil with some 
organic solvent such as xylene hexafluoride or 
diethylbenzene, it cannot be assumed that the 
bonds will remain unaffected by the immersion 
liquid. In fact, it has been reported^® that Vulca¬ 
lock bonds are unsatisfactory in a liquid con¬ 
taining 85 per cent DB castor oil and 15 per cent 
diethylbenzene. 


^ ^ Biityl-C Cement 

Most cements that have been investigated do 
not permit a transducer to be operated at very 
low temperatures owing to the difference in the 
coefficient of thermal expansion between the 
crystals and the rigid base to which they are 
customarily cemented. This usually results in 
fracturing the crystals long before a tempera¬ 
ture of —40 C is reached. Since there is a tend¬ 
ency to extend transducer specifications to in¬ 
clude —40 C as the lower operating limit of 
temperature, an effort has been made to secure 
a satisfactory cement for this purpose. As the 
result of an extensive investigation at BTL,^*’’ 
Butyl-C cement has been found to fulfill this 
specification. The photograph in Figure 43 






310 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


shows an ADP crystal cemented to a Durez 
wafer with Butyl-C cement, and the Durez 
wafer in turn cemented to a steel block with 
XCU 16257 urea formaldehyde cement. It will 
be noted that no fractures have appeared in 
this crystal, even though it has been cooled 
down to —40 C in a fairly short time interval. 
The same is essentially true of another ADP 
crystal, illustrated in Figure 44, which has 
been attached to a porous ceramic wafer with 
Butyl-C cement, the other conditions being 
practically identical. It was found in this case 
that a very small crack appeared at —39 C. For 
comparison, reference should be made to Fig¬ 
ures 41 and 42 where ADP crystals have been 
bonded with Vulcalock cement and subjected to 
the same treatment as just described. The su¬ 
periority of Butyl-C cement for low-tempera¬ 
ture applications will be immediately evident. 

In concluding their report, Frosch and Wil¬ 
liams^® recommended the use of Butyl-C cement 
for both ADP and RS where there was danger 
of cracking at low temperatures. They found 
that it was necessary to employ porous-ceramic 
insulators rather than Durez where high Q val¬ 
ues and high power were required. A rigid bond 
between these ceramic wafers and steel resona¬ 
tors was also necessary. Butyl-C cement can be 
used safely in contact with DB castor oil al¬ 
though other immersion liquids cannot be rec¬ 
ommended without further tests. 

It is regretted that more definite information 
on the composition of Butyl-C cement is not at 
hand. The principal ingredient is a polymer 
which is composed of a curing synthetic rubber 
modified with an aliphatic thermoplastic resin. 
In manufacturing this cement it is originally 
prepared in two parts, A and B, according to 
the following directions. 

Part A 

100 parts by weight polymer 
5 parts by weight zinc oxide 
3 parts by weight stearic acid 
V 2 part by weight sulphur 
2 parts by weight GMF 

The above constituents, with the exception of the 
GMF, are thoroughly mixed on cold differential mixing 
rolls. After complete mixing is obtained, the GMF is 
added and mixing continued for as short a time as 
possible. We have found that the material should not 
be allowed to heat appreciably during the mixing as 


the molecular weight of the polymer is reduced or the 
GMF causes gelation at elevated temperatures. 

Part B 

100 parts by weight polymer 

5 parts by weight zinc oxide 

3 parts by weight stearic acid 
IV 2 parts by weight sulphur 

4 parts by weight lead peroxide (PbOo) 

These components are mixed thoroughly on cold 
differential mixing rolls. 

Both Part A and Part B are made up into a 30 per 
cent solution in benzene. To each 700 cc of mixed 
cement is added 21 cc of isopropyl alcohol. 

Butyl-C cement is ready for use when equal 
parts by volume of components A and B are 
thoroughly mixed. The mixture has a useful 
life of about 2 hr. When cementing metal foils 
to crystals the Butyl-C mixture is further di¬ 
luted with an equal volume of benzene. In using 
Butyl-C cement a thin brush coat is applied to 
the required areas and the cement is allowed to 
dry until its surface becomes dull in appearance. 
The surfaces which are to be cemented together 
are then assembled in an appropriate jig with 
a pressure of 6 to 7 psi. The jig is placed in an 
oven at 60 C and the assembly allowed to cure 
for 24 hr. 

Butyl-C cement was developed originally by 
the plastics group of the Chemical Department 
of BTL at Murray Hill, New Jersey. Some diffi¬ 
culty has been experienced in obtaining a pre¬ 
pared cement from commercial sources which 
would duplicate the original material satisfac¬ 
torily, It is regretted that the exact composition 
of the polymer entering into the manufacture 
of Butyl-C cement is not available to the writer 
for inclusion in this volume. Further informa¬ 
tion regarding it can be obtained from BTL. 


® ^ Bonding ADP Crystals to Rubber 

The bonding of piezoelectric crystals to rub¬ 
ber has proved to be a valuable technique. It 
has given rise to the development of one type 
of inertia drive transducer at UCDWR in which 
the radiating face of the crystal is bonded di¬ 
rectly to the rubber window of the case. In 
another type, the nonradiating end of the crys¬ 
tal is bonded to a thin supporting strip of rub¬ 
ber which is later backed by a pressure release 




CEMENTS 


311 



































312 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


of cellular rubber as illustrated in Figures 57 
and 58. Crystal arrays bonded to strips of rub¬ 
ber in this manner are readily formed into vari¬ 
ous configurations as discussed in Section 8.7.3. 

A standard procedure in bonding many mate¬ 
rials to rubber is to make direct use of Type 
55-6 Cycle-Weld cement, a trademarked prod¬ 
uct of the Chrysler Corporation, Cycle-Weld 
Division, Detroit, but this method is not appli¬ 
cable with ADP crystals inasmuch as Type 55-6 
cement does not bond well to ADP. The tech¬ 
nique that has been developed at UCDWR con¬ 
sists in the application and curing of a priming 
coat of Type 55-6 cement on the rubber, the 
crystals being bonded later to this priming coat 
with Type C-3 Cycle-Weld cement. The steps in¬ 
volved in this process are portrayed in Figure 
45 and a sample of the processing record sheet 
found convenient at UCDWR is reproduced in 
Figure 46. 

Rubber or neoprene is first cleaned thor¬ 
oughly to remove talc and any other contami¬ 
nating substance; sanding may be necessary 
with some samples of sheet material. The sur¬ 
face to be bonded is then cyclized by covering 
it with, or immersing it in, concentrated sul¬ 
furic acid for from 3 to 15 min (Figure 45A). 
Too long a period results in a brittle surface 
layer of appreciable thickness so it is better to 
try first the lower time limit stated above on any 
given type of material. For neoprene, 3 to 5 min 
has been satisfactory; for pC rubber, 3 min or 
less. After washing off the excess acid with a 
generous amount of water, place the rubber in 
a tray with running water for perhaps an hour 
in order to insure the removal of the acid 
(Figure 45B). Then wipe the rubber dry with 
clean toweling and warm gently to insure that 
the surface is moisture free. It is very impor¬ 
tant that the sulfuric acid be thoroughly re¬ 
moved since its hygroscopic nature would result 
in a water layer being formed on the surface of 
the rubber. This would be detrimental to the 
bond as well as to the electrical resistance of 
the crystal. 

The Type 55-6 Cycle-Weld cement is applied 
to the prepared rubber surface with a fine 
camel’s hair brush (Figure 45C). It should be 
brushed as quickly and evenly as possible over 
the surface. It is difficult to do this without 


leaving brush marks owing to the rapid evapo¬ 
ration of the solvent (methyl ethyl ketone). If 
the thickness of the priming coat permits, the 
ridges which result from uneven brushing may 
be sanded off following the curing process. 
Spraying of the Type 55-6 cement has not been 
satisfactory and is not recommended. The prim¬ 
ing coat is allowed to dry at least 30 min at 
room temperature and then 10 min at 80 C. The 
object of these two steps is the slow removal of 
the solvent; an alternative procedure is to hold 
the samples at room temperature for 4 to 48 hr. 
To bring about the thermosetting of the Type 
55-6 cement, it should be cured for a period of 
at least 60 min at 125 C. At 150 C, 15 min should 
be sufficient. 

Type C-3 is a thermosetting adhesive for 
bonding metals, wood, glass, and plastic ma¬ 
terials. It adheres well to ADP but not to rub¬ 
ber. However, it makes an excellent bond to the 
previously cured priming coat of Type 55-6 
cement. After cleaning the crystal surface by 
wiping with cheesecloth moistened with methyl 
ethyl ketone or other suitable solvent, the C-3 
cement is applied with a camel’s hair brush. 
Figure 45E, or a drop is added with a small 
wood stick and then spread evenly over the area. 
To secure a layer of cement 0.0015 to 0.002 in. 
thick, one must make a liberal application of the 
liquid and spread it out very quickly. If an at¬ 
tempt is made to brush it thinly over the sur¬ 
face of a crystal, it is very likely to streak. Any 
further application of cement will redissolve the 
original layer thus giving a streaked or spotty 
film of variable thickness. A little practice will 
soon teach one an acceptable technique. The 
manufacturer recommends spraying and sup¬ 
plies for this purpose a special spray cement, 
but thinning the brush-type cement with methyl 
ethyl ketone is also satisfactory. Extensive 
“cobwebbing” is encountered with too thick ce¬ 
ment. In production work, an effort should cer¬ 
tainly be made to master a satisfactory spray 
procedure. 

In the same manner (Figure 45D), Type C-3 
cement is brushed or sprayed over the Type 55-6 
priming coat which has been cured previously 
on the rubber. 

The C-3 cement applied to the crystals and 
to the rubber is now allowed to dry at room tern- 



CEMENTS 


313 


CYCLE-WELD PROCESSING RECORD 

Technician ___ Date _ 

Type of Rubber _ Material _ Transducer No. _ 

A. Apply concentrated sulfuric acid to rubber surface for 3 min. 

Time in : Time out : Elapsed time = : 

B. Wash rubber surface for 30 min. in running water, then rinse 
in distilled water, air-dry thoroughly and then oven-dry at 
1500 F for 10 minutes. 

Time in ; Time out ; Elapsed time = ; 


c. 

Brush 55-6 cement 

on rubber, then dry 

and cure as 

follows; 

i 


Room temperature 

(30 min.); from 

• 

to : 

= 

min. 


180^ oven 

(10 " ); " 


^0 _L_ 


min. 


270°F " 

(60 " ); " 

• 

to : 

, = 

min. 

D. 

Brush C-3 cement on rubber (apply over 

the cured 

55-6 cement). 


Room temperature 

(30 min.); from 

• 

to : 

= 

min. 


180 °F oven 

(25 " ); " 

• 

to 

= 

min. 

E. 

Brush C-3 on ADP crystals (cleaned 
ketone) and dry at: 

previously with methyl 

ethyl 


Room temperature 

(30 min.); from 

• 

• 

to : 

= 

min. 


180 

(25 » ); 

• 

to : 

= 

min. 


F. Assemble in press, apply 15-25<lbs/in^ pressure and cure at 
270 OF for 90 minutes at glue line . 

Time in _ • _ Time out ; = min. 

REMARKS; (Write below any unusual behavior or any irregu¬ 
larities in processing.) 

Figure 46. Cycle-Weld processing record used at UCDWR. 
































314 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


perature for at least 30 min in order to permit 
an initially low rate of escape for the solvent. 
Then it may be heated 25 min at 80 C to re¬ 
move the solvent completely. An alternative 
procedure consists in drying at room tempera¬ 
ture only, but for a period of 2 to 48 hr. 

The surfaces to be bonded are assembled in 
their final position (Figure 45F), and a pres¬ 
sure of 25 to 60 psi is applied. The exact pres¬ 
sure used depends on the area involved and on 
the hardness of the rubber; this value can be 
estimated visually, using as a criterion the ab¬ 
sence of any marked distortion of the rubber. 
To insure a uniform pressure, a pneumatic 
press, such as illustrated in Figure 45G, is 
recommended. 

To cure the C-3 thermosetting cement, the 
assembly is heated to 125 C in an oven and 
maintained with this temperature at the glue 
line for at least an hour. Although the timing is 
not very critical, with a longer period probably 
being beneficial, the damaging of the ADP crys¬ 
tal surfaces by prolonged exposure to heat con¬ 
stitutes the limiting factor. Too long exposures 
of ADP to temperatures of 125 C, and especially 
to 150 C, causes a surface breakdown of the 
ADP with the emission of ammonia vapor and 
the appearance of a phosphoric acid layer on the 
crystal. 

ADP crystals must be allowed to cool slowly 
down to at least 80 C to avoid fracture from 
thermal strains. In the laboratory, it is often 
convenient to turn off the power and allow them 
to cool overnight without removal from the 
oven; in production, an annealing oven would 
be used. 

If the outlined procedure is correctly fol¬ 
lowed, the resultant bond should be stronger 
than either the rubber or the crystal. Tests of 
these bonds usually ruptured the rubber; only 
occasionally did a crystal break. For maximum 
bond strength, the rubber used should have high 
tensile strength, low free sulphur content, low 
percentage of mineral filler, a hardness of 40-70 
Shore durometer, and should be resistant to the 
curing temperature. 

High humidity is detrimental to this type of 
bond, apparently by its effect on the C-3 cement 
layer. Provision should be made for the inclu¬ 
sion of a drying agent, such as silica gel, in any 


transducers employing this ADP-rubber con¬ 
struction. Extended field trials have not yet 
been made. Reference may be made elsewhere 
in this volume for a discussion of transducers 
which have been designed to take advantage of 
this technique. 


Thermoplastic Cements 

Although little use has been made of thermo¬ 
plastic cements in the construction of crystal 
transducers, they would seem to offer good pos¬ 
sibilities in this direction. This is particularly 
the case for ADP crystals since they are quite 
capable of withstanding the required high tem¬ 
perature. The only thermoplastic cement em¬ 
ployed to date with ADP crystals has been a 
modified polyvinyl acetate which has been used 
in the manufacture of spliced crystals as 
pointed out in Section 8.3.7. 

From experience of a preliminary sort at 
UCDWR, Butacite VF-7100 cement, an unplas¬ 
ticized polyvinyl butyral produced by Du Pont, 
appears to have much promise for crystal appli¬ 
cations. In bonding ADP crystals to each other 
and to steel very high Q joints were obtained 
with it. The tests were not carried out over a 
sufficiently long period to test their endurance 
under various conditions so that no specific 
recommendation can be made. In the opinion of 
the writer further investigations should be con¬ 
ducted with this cement. 

It is recognized that almost every manufac¬ 
turer of plastics produces one or more types of 
adhesive in the thermoplastic category. There 
is neither intention nor sufficient basis to indi¬ 
cate the superiority of any particular type or 
brand for crystal applications at this juncture 
but merely to point out that in the limited ex¬ 
perience at UCDWR one or two types have been 
tried and found promising. Since the bonding 
of crystals to supporting structures is a very 
specialized application, it is perhaps not to be 
expected that existing compositions of thermo¬ 
plastic cements will be entirely suitable. It is 
more realistic to assume that variations in com¬ 
position must be systematically investigated 
with a view to obtaining the necessary and de¬ 
sirable qualities for each specific type of bond- 




CEMENTS 


315 


ing operation. The qualities that must be con¬ 
sidered have been discussed explicitly in Sec¬ 
tion 8.6.1 and implicitly in Section 8,6.2 and 
succeeding sections. 

Cements Containing Catalysts 
Urea Formaldehyde 

In conducting tests on ADP crystals bonded 
to wafers and then to steel plates, BTL has 
made use of a catalyzed urea formaldehyde ad¬ 
hesive.^® These tests have been referred to in 
Sections 8.6.4 and 8.6.5. The particular adhesive 
employed was a product of the Bakelite Cor¬ 
poration and consisted of a liquid, XCU 16257, 
and a solid catalyst, XK 16229. These two ma¬ 
terials were mixed thoroughly just before using 
and in the ratio of 10 g of the liquid to 0.8 g of 
the catalyst. This mixture has a usable cement¬ 
ing life of approximately 2 hr. 

For the tests mentioned, a thin brush coat of 
the cement mixture was applied to the steel and 
to the insulators, and the parts allowed to cure 
unassembled overnight at room temperature. 
After rough lapping the cement surface of both 
the steel and the insulators, a second coat of 
cement was applied. The insulators were then 
assembled onto the steel plates and sufficient 
pressure was applied to hold the parts in place. 
The assemblies were cured for 24 hr at room 
temperature. 

Norace Cement 

Norace cement, a product of the Norton Com¬ 
pany, Worcester, Massachusetts, is a thermo¬ 
setting plastic which sets at room temperature. 
It has been used for cementing ADP bars to 
supporting plates by BTL, as discussed in Sec¬ 
tion 8.3.4. For this application the method of 
preparation of the cement has been described 
as follows.'^ Ten cubic centimeters of the powder 
was measured out and emptied into a wax- 
paper cone. This cone was made by folding the 
paper as in a chemical filter and was then held 
in a 60-degree conical depression in a lead block. 
A stirring rod was used to make a depression in 
the powder and 3 cc of acidified solvent were 
added and quickly stirred to form the cement of 
mud-like consistency. The paper cone was then 
removed from the lead block so that most of the 


material could be scraped from it and trans¬ 
ferred to the mounting block. The acidified sol¬ 
vent was prepared by adding 5 per cent of gla¬ 
cial acetic acid to the number 1 solvent fur¬ 
nished with the cement. 

® Miscellaneous Adhesives 

Molten Rochelle Salt 

In an early studyon glued joints and the 
acoustic losses which occur in them, it was 
found that fused RS was one of the four most 
promising cements investigated. In fact, fused 
RS resulted in the hardest and most loss-free 
bond. This cement is rather difficult to use and, 
if the crystal is cemented to a support having a 
different coefficient of expansion, it is liable to 
crack when used over an extended temperature 
range. Perhaps its most satisfactory application 
is in bonding RS to RS, Two very common cases 
arise which call for a RS to RS bond. One is the 
production of spliced crystals, already discussed 
at some length in Section 8.3,7, and the other 
is in the production of bimorphs. 

Clear fragments of RS may be fused by plac¬ 
ing them in a clean utensil (stainless steel is 
quite satisfactory) and heating slowly to a 
temperature of 93 C. It may be advantageous to 
use a water bath or double-boiler arrangement. 
It is important that everything be kept meticu¬ 
lously clean and that not over a few hours 
supply of material be made up at one time. 
While awaiting use, the fused RS should be 
maintained at 93 C in a melting pot and dipped 
out in small quantities from time to time. At 
the Brush Development Company, where this 
cement is used in the commercial production of 
bimorphs, the fused RS is ladled onto stainless 
steel slabs 5x8x% in. thick in order to permit 
some precooling before applying the cement to 
the crystals. After skimming the surface of the 
cement pool on the slab with a wooden stick, the 
face of the crystal to be cemented is dipped into 
the pool of fused RS and then placed on the 
other half of the bimorph with a gentle sliding 
pressure. 

Acryloid B7 

The second promising cement referred to in 
the discussion in Section 8.6.9 is Acryloid B7, 





316 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


a product of Rohm and Haas. According to the 
study^“ mentioned, it appears to give a better 
bond to a smooth surface, such as steel, than 
some other cements. One disadvantage of Acry- 
loid B7 is that its drying time is somewhat 
longer than for Vulcalock or bakelite cement. 
However, this defect might be improved by the 
use of a different solvent. 

The use of Acryloid B7 at UCDWR has been 
limited almost exclusively to the cementing of 
tin-foil electrodes to crystals, according to the 
steps outlined in Section 8.3.8. 


« " PREPARATION OF ARRAYS 
“ Layout and Assembly 

Crystal arrays for transducers are designed 
with so many variations that it is extremely dif¬ 
ficult to describe satisfactory methods of assem¬ 
bly that would be generally applicable. It should 
be pointed out that perhaps too much stress 
cannot be placed on the necessity for appropri¬ 
ate jigs and auxiliary devices. Not only will 
time and effort be saved in the final assembly 
operations, but the results achieved will be 
much more uniform. The designing of special 
jigs is a problem that must depend for its solu¬ 
tion on the ingenuity of the shop foreman or of 
the transducer designer. How elaborate a par¬ 
ticular jig should be will probably depend on 
whether it is made for production work or 
merely for a few transducers of a given design. 

In this section it will be the intention to in¬ 
dicate how various simple crystal arrays may be 
assembled by means of satisfactory jigs and 
how these assembled arrays may be mounted on 
permanent supports. With one or two excep¬ 
tions, subject material and illustrations for this 
section have been based on experience at the 
UCDWR Transducer Laboratory. 

Simple Flat Arrays 

Full-scale drawings of the crystal array are 
usually made available to the construction fore¬ 
man by the designer. The crystals, when fur¬ 
nished previously with electrodes, may be ar¬ 
ranged on a glass plate according to the design 
drawing. It is necessary to pay attention at this 


point to the polarity marks in order to make 
sure that all the crystals are properly oriented. 
For a simple array where the crystals are all 
driven at a single velocity, the polarity arrange¬ 
ment is usually as shown in Figure 56. 

Where individual crystals have previously 
been furnished with electrodes which do not 
permit a direct soldering of wires to them, it 
will often be found convenient to assemble the 
crystals into linear strips or bars. This type of 
assembly is illustrated in Figure 47 where a 
row of crystals that has been cemented to a 



Figure 47. A strip of crystals, which have been 
bonded previously to a foil, is being placed in 
position in a simple jig as one step in the assem¬ 
bly of an array. 

narrow strip of silver foil is shown being placed 
in position in a simple jig. In order to insure 
correct alignment of the crystals while cement¬ 
ing them to the foil in such a row, the edge of 
each crystal was allowed to rest against a guide. 
The separation distance between adjacent crys¬ 
tals along the strip is adjusted with a spacer 
each time an additional crystal is added. As 
each strip of crystals is placed in the jig, care 
must be taken to observe polarity requirements. 
In an array of this type, the individual rows of 
crystals may or may not be bonded to each 
other. As illustrated in Figure 47 the rows are 
not bonded together. After all the strips are as¬ 
sembled in the jig, a thin layer of compliant ma¬ 
terial should be placed as a facing against both 
of the metal bars which act as clamps. The 
metal bars are then pulled together by tighten- 










PREPARATION OF ARRAYS 


317 


ing the nuts on the threaded rods. Before a final 
tightening, the assembly of crystals may be ad¬ 
justed for correct alignment by the use of a 
common carpenter’s square. When a small unit 
has been completely assembled and tightened, 
the entire jig may be lifted without the crystals 
being displaced. Planeness of the array is ob¬ 
tained by having all of the crystals lying on a 
piece of plate glass or on a surface plate before 
clamping. Owing to irregularities in the crys¬ 
tals, the surface of the array may still not be 
sufficiently flat so that a subsequent grinding 
operation may be required. This is particularly 
true when the array is to be attached to flat or 
rigid plates and will be discussed further in 
Section 8.7.2. 

In arrays where the electrodes consist of 
heavy foil which is sufficiently thick to permit 



Figure 48. A flat crystal array clamped in 
position in a simple jig, ready to be attached 
to a backing plate. 


wires to be soldered directly to them, the crys¬ 
tals may be bonded to long strips of foil which 
cover the entire electrode area of each crystal 
face. Such an array is shown in Figure 48 and 
it will be noted that the foils extend an appre¬ 
ciable distance beyond the end of the crystals. 
Any excess length of foil may be trimmed off 
after the foils are wired as discussed in Section 
8.7.4 and illustrated in Figure 61. It will also 
be observed that foils of alternate polarity ex¬ 
tend on opposite sides of the array. The polarity 
arrangement is as shown for the array of Fig¬ 
ure 56. 

In simple arrays where the crystals are not 
intended to be in contact with each other but 
are spaced individually as in Figure 57, or in 
groups of a few crystals each as in Figure 49, 
provision must be made for maintaining the 
spacings while the crystals are being bonded to 
supporting structures. Bakelite or micarta sepa¬ 
rators have been found convenient for the main¬ 


tenance of such spatial arrangements (see Fig¬ 
ure 55). To insure that the plastic strips may 
be readily removed following the final cement¬ 
ing operation, it is advisable to cover each in¬ 
dividual strip with a layer of plain paper. After 
the crystals have been bonded permanently to 
their support, the paper-covered separators may 
be readily removed from between the crystals. 

Lobe-Suppressed Flat Arrays 

The layout and assembly of flat arrays in¬ 
volving some scheme of lobe suppression usually 
presents more difficulties than an array driven 
at a single voltage. A common type of lobe sup¬ 
pression array is shown in the retouched photo- 



Figure 49. Photograph of a UCDWR split 
array which has been retouched to exaggerate 
the foiling arrangement for a 3 to 1 scheme of 
lobe suppression. The crystals in the central 
region operate at full amplitude; in the periph¬ 
eral region, at Vs amplitude. The two halves of 
the array may be operated either in phase or 
out of phase. 

graph of Figure 49 for a transducer where it is 
intended that the two halves of the crystal 
motor may be operated either in phase or out of 
phase. The crystals in the central part of this 
array are driven at three times the velocity of 
the crystals in the peripheral region. This ratio 






318 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


of driving voltage is provided by having all of 
the crystals in the central region in parallel, 
while in the outer zone, the same driving volt¬ 
age is applied to each group of three crystals 
connected in series. The simplicity of wiring 
arrangements for this 3 to 1 ratio will be appar¬ 
ent after studying Figure 49 in which the ap- 



Figure 50. The lobe-suppressed crystal motor 
of the QBF echo-ranging projector. The central 
zone consists of 24 full amplitude blocks of 4 
crystals each; the outer zone 28 half amplitude 
blocks of 2 crystals each. (Bell Telephone 
Laboratories.) 

pearance of the foils has been exaggerated for 
emphasis. It will be noted that every third foil 
is continuous past the crystals of both the inner 
and outer zones. The triplet crystal groups in 
the outer zone may, of course, be replaced by 
single crystals having the same total thickness. 
Some gain in uniformity of the electrical field 
in these crystal groups could be obtained by fur¬ 
nishing intermediate electrodes between the in¬ 
dividual crystals of each group. In general, 
however, this procedure has not been followed 
at UCDWR. 

In constructing the array of Figure 49 the 
triplet groups were first bonded together with 
bakelite cement. Complete half rows were next 
assembled by starting with a long silver foil of 
the correct width and successively cementing 
crystals, triplet blocks, and foil strips to it, with 
the proper spacing and polarity orientation, 
until the 11 pairs of semirows were prepared. 
These assembled rows were then placed in a 
pneumatic press and cured according to the 


schedule for bakelite cement given in Section 
8.6.4. After curing the cement bonds in this 
manner, the rows of crystals were assembled in 
a jig with the proper spacers to give the con¬ 
figuration shown in the figure. The array was 
then mounted on its backing plate as directed in 
Section 8.7.2. 

A transducer employing a 2-to-l type of lobe 
suppression is shown in Figure 50. This type of 
crystal motor is used with the QBF echo-rang¬ 
ing system projector and was designed by BTL. 
The same driving voltage is applied to each and 
every crystal in this array but owing to the fact 
that the crystals in the central zone are just 
half as thick as the crystals in the peripheral 
zone, the crystals in the central zone are driven 
with twice the velocity. In other words, we have 
what is known as a 2-to-l scheme of lobe sup¬ 
pression. It will be noted that the crystals are 
bonded together in groups, each group having a 
radiating face one inch on a side. In the central 
zone four crystals comprise a group; in the 
outer zone, two crystals. In this transducer the 
wiring connections to the evaporated gold elec¬ 
trodes are made with strips of gold-plated nickel 
silver foil 0.001 in. thick. A close inspection of 
the photograph in Figure 50 will reveal dark 
areas where these foils come in contact with the 
interconnecting wires. The spacing between the 
crystal groups is % in. in both directions. 

Cylindrical and Curved Arrays 

Several possibilities exist for the mode of as¬ 
sembly of cylindrical or curved arrays. One type 
occurs in the crystal motor which is used as the 
sound source in the UCDWR-type CQ trans¬ 
ducer and is illustrated in the photograph of 
Figure 33 of Chapter 1. In this case, quartets 
of crystals are bonded together and seven of the 
quartet groups are attached to each backing 
bar, the bars themselves being so arranged as 
to constitute part of a cylindrical surface. In a 
second type, illustrated photographically in Fig¬ 
ure 40 of Chapter 1, the crystals are bonded 
directly to the interior of a rubber cylinder. A 
jig for the assembly of the crystals for this type 
transducer is shown in Figure 51. The crystals 
are foiled on both sides in linear strips of four 
crystals each. The foils, of 0.0015-in. silver 
sheet, extend past each group of crystals at one 













PREPARATION OF ARRAYS 


319 


end or the other in order to provide soldering 
lugs. These crystal strips are prepared and 
cured in advance of the final assembly opera¬ 
tion, at which time they are inserted in the 
radial slots of the jig illustrated in Figure 51. 
When all of the strips of crystals are in place, 
the jig is then lowered into a reinforced rubber 
cylinder, shown in Figure 68. When finally ad¬ 
justed to their correct position, so that the crys¬ 
tals are located midway between the steel rods 
in each case, the rubber tube within the jig is 



Figure 51. A special jig used at UCDWR for 
assembling the cylindrical inertia-drive trans¬ 
ducer shown in Figure 40 of Chapter 1. By 
inflating the rubber core of the jig the radiating 
faces of the crystals are pressed firmly against 
the interior surface of a reinforced rubber 
cylinder (see Figure 68) during the bonding 
operation. 

inflated, thus forcing the radiating face of each 
crystal out against the interior wall of the rub¬ 
ber cylinder. The bonding to the rubber is done 
by means of the technique described in Section 

8 . 6 . 6 . 

Another method for the production of cylin¬ 
drical arrays will be made clear by an inspection 
of Figures 57, 58, and 62. The crystals in this 
case possess independent silver foils with sol¬ 
dering lugs and are bonded to a thin flat sheet of 
rubber. Following the bonding process, accord¬ 
ing to the procedure outlined in Section 8.6.6, 
the crystal assembly is coiled about a circular 
support such as illustrated in Figure 58 or 62. 
They are held in place by a combination of 
wrapping with nylon thread and cementing to 
a central core. 


In assembling cylindrical arrays, it is very 
important to observe the polarity marks on the 
crystals. If alternate pairs of foils are to be 
positive and negative respectively, an even num¬ 
ber of crystals or crystal groups must be used 
around the circumference of the cylinder. 

Stacked Arrays 

Perhaps the least troublesome type of array 
to assemble is that of a simple stack. Each indi¬ 
vidual crystal is first furnished with electrodes. 
Where the electrodes are of a type that do not 
possess lugs for soldering, narrow foil strips 
are cemented lightly to each crystal and brought 
out either at the ends or sides of the crystals 
as shown in the illustrations in Figure 59 and 
Figure 60. Where the electrodes do possess sol¬ 
dering lugs, a single-foil electrode may be used 
between each pair of crystals. Whether the en¬ 
tire stack of crystals should be bonded into a 
single block is questionable from the standpoint 
of efficiency. Jigs for maintaining the correct 
alignment in such a crystal stack can obviously 
be of a very simple type and no illustrations of 
jigs are included here. 

The stacks may be arranged in various ways 
to meet design specifications. The crystals may 
radiate either off their end faces or off their 
side faces. They may be arranged spirally as in 
Figure 38 of Chapter 1, or make various angles 
with each other. Most of the stack-type trans¬ 
ducers made at UCDWR have been assembled 
from RS crystals and resemble Figure 59. Non¬ 
radiating faces were blanked off with one of the 
isolating materials discussed in Section 8.7.5. 

“ ■ “ Backing Plates 

Steel Backing Plates 

Steel has been used more frequently than any 
other material as a backing plate for crystal ar¬ 
rays. The choice of steel has been based in part 
on its mechanical strength and on its machina- 
bility as well as on its acoustical behavior. In 
using a conducting backing plate, provision 
must be made at the outset for sufficient elec¬ 
trical insulation to withstand the voltages em¬ 
ployed. This insulation may be provided in a 
variety of ways, all of which may be entirely 
satisfactory. 



320 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


The customary method at UCDWR has been 
to apply a thick coating of porcelain enamel di¬ 
rectly to the steel. The porcelain enamel base 
used for this purpose consists of a low melting 
point glass frit and is applied to the steel by the 
so-called dry process. In this dry process the 
backing plate is first heated in a furnace to a 
cherry-red color and the frit is sprinkled on the 
surface by means of a dusting screen. The frit 
thus deposited immediately melts and so forms 
a glazed surface. This layer should be smooth 
and free from bubbles if properly applied. How¬ 
ever, a few small bubbles usually exist in such 
porcelain layers and may later give rise to elec¬ 
trical breakdown. A method for treating the 
porcelain in order to prevent voltage breakdown 
will be discussed in a later paragraph. 

The surface of these porcelain coatings is 
never sufficiently flat for attaching large crystal 
arrays. In addition to slight irregularities on 
the surface, there is also a rounded edge or a 
meniscus at the border between the porcelain 
and the steel. Grinding or lapping of the sur¬ 
face is therefore necessary. This surfacing may 
be accomplished by first grinding with coarse 
silicon carbide and then finishing with a finer 
grade of silicon carbide. Suitable grades for this 
purpose are No. 60 and No. 80. A standard lap¬ 
ping technique is to use a large brass platen or 
grinding flat with the backing plate itself con¬ 
stituting a tool. The flatness of the resulting 
surface obviously depends on the skill of the 
operator, but there should be little difficulty in 
attaining a surface which is flat within ±0.003 
in. 

In cases where holes must be drilled through 
the backing plate, the drilling should be done 
before the porcelain coating is applied, other¬ 
wise there is danger of cracking the porcelain. 
Should it prove necessary to provide holes 
through the porcelain layer, it can be done by 
grinding with a tool designed for this purpose 
while employing wet silicon carbide as an abra¬ 
sive. Wood dowels make satisfactory tools. In 
some instances it has proved desirable to divide 
the surface of a backing plate into two or more 
areas in order to minimize vibration or to segre¬ 
gate regions of a plate underlying crystal 
groups driven at different velocities for pur¬ 
poses of lobe suppression. In such cases it is 


likewise essential to wet-grind the porcelain 
with specially designed tools employing silicon 
carbide. 

After the glazed coating of the porcelain has 
been removed by grinding, a fine porosity will 
be evident together with perhaps a few larger 
holes which are plainly visible. If these holes 
which have resulted from grinding into fairly 
large bubbles originally present in the porce¬ 
lain remain, voltage breakdown may occur at 
perhaps 1,000 v or less. By using a leak tester, 
such as employed for detecting leaks in glass 
vacuum systems, all defective spots should be 
located and marked. 

In order to improve the breakdown voltage of 
porcelain-coated vacuum plates, the following 
procedure is currently employed at UCDWR. 
The porcelain is first cleaned with benzine 
and then scrubbed thoroughly with Glyptal 
thinner No. 1500. The backing plate is then 
warmed to about 120 F and a thin layer of clear 
Glyptal is brushed over the porcelain surface. 
The plate is now placed in a vacuum chamber 
where a low pressure is established and then 
broken two or three times. Finally, the plate 
should be left in the evacuated chamber for a 
period of time sufficiently long to enable the 
plate to cool to room temperature. This may re¬ 
quire 2 or 3 hr. Upon removal of the plate from 
the vacuum chamber, the excess Glyptal should 
be scraped off with a razor blade. After allowing 
the plate to set for an additional 2 hr it is 
sanded freely with a fine grade of silicon car¬ 
bide paper (Carborundum 220A-320A). After 
allowing the plate to set again for an hour or 
more, the porcelain surface should be washed 
with a cheesecloth dampened with benzene. The 
porcelain should be voltage checked again with 
the leak tester. If satisfactory, it is ready for 
use; if not, it must be treated again in a similar 
fashion in order to fill up all defective cavities 
with Glyptal. 

A second method of providing insulation in 
metal backing plates consists in cementing in¬ 
sulating wafers between the crystals and the 
backing plate. The Bell Telephone Laboratories 
have favored this method, having used origi¬ 
nally a ceramic wafer approximately in. 
thick. These wafers contained narrow flutes or 
channels every % in. to permit the escape of 




PREPARATION OF ARRAYS 


321 


excess cement and solvent vapor. Later, prefer¬ 
ence was given to a plastic wafer made of a 
Durez resin. These wafers were cemented to the 
steel with a very hard cement in which the 
acoustic losses were reduced to a very low value 
(see Figures 41 to 44). 

The Naval Research Laboratory has made 
use of a 344 -in. bakelite material as an insulator. 
This product was similar to micarta but con¬ 
tained a cloth insert. 

Glass and Plastic Plates 

By resorting to glass backing plates, one 
avoids completely the intermediate insulating 
materials required for any electrically conduct¬ 
ing backing plate. This appears to be a real ad¬ 
vantage in that it avoids some energy loss which 
takes place in the additional cement layer. It 
also reduces stray electrical capacitance to the 
backing plate. A ground surface on the glass is 
probably advisable in order to obtain improved 
adhesion to the crystals. This ground surface 
may be made by using 80-grit silicon carbide 
as an abrasive. 

Care should be taken to see that all glass used 
for backing plates is well annealed and without 
strain. Strains can be readily detected with 
polarized light by means of the device shown 
in Figure 33. Mounting of glass backing plates 
offers somewhat more of a problem than metal, 
especially where holes are to be provided. Al¬ 
though holes may be readily drilled in glass 
plates, they do reduce the strength at that point 
and increase the likelihood of breakage. A more 
desirable method for mounting is to bevel the 
edges of the plate in such a manner that suitable 
wedges may be used to hold the backing plate 
in position, preferably in a shock-absorbent 
mounting. 

Miscellaneous Plates 

Metal backing plates other than steel have 
been found useful, particularly for low-fre¬ 
quency applications where the thickness of a 
steel backing plate becomes excessive. Lead has 
found the most extensive use. Since the velocity 
of sound in lead is about one-fourth that of 
steel or glass, lead backing plates are thinner by 
a ratio of approximately 4 to 1. Consequently, 
there is a large resultant saving in both space 


and weight. Lead is most conveniently used as 
a backing plate by adding it to a rather thin 
steel plate which has already been porcelain 
coated. Since the melting point of lead is 328 C, 
difficulty may be experienced in coating the 
steel with lead without cracking the porcelain. 

Where other types of insulation are em¬ 
ployed, namely, ceramic or porcelain wafers or 



Figure 52. A unit type of backing-plate con¬ 
struction developed at the Naval Research Labo¬ 
ratory, The thick black rubber washer forms an 
air seal for the metal cap and provides a shock 
mounting for the backing plate. 

thin bakelite sheets, there would seem to be no 
objection to attaching these directly to a lead 
plate, providing the lead plate is capable of sup¬ 
porting itself against mechanical deformation. 

The low melting point alloy called Cerrobend 
may prove useful in place of lead for backing 
plates. A porcelainized steel plate of sufficient 
thickness is first tinned properly and then the 
Cerrobend is poured on the tinned surface until 
the retaining mold is filled to the desired depth. 
Since Cerrobend melts at a temperature below 
the boiling point of water, the operation may be 
carried out safely without cracking the porce¬ 
lainized surface. The final operation is to chuck 
the backing plate in a lathe and turn off the ex- 





322 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


cess metal. Any holes for the accommodation of 
electric leads or for oil filling may be provided 
at the time of casting the Cerrobend by insert¬ 
ing suitable cores at the desired locations. 

Duralumin has been used in the unit-type 
backing-plate construction originating at NRL. 
This construction is shown in Figure 52 where 
it will be noted that the backing plate consists 


The crystal block is also surrounded on all four 
sides by the same type of bakelite material. 

Mounting Technique 

In cementing flat arrays of crystals to back¬ 
ing plates, it is essential that both the backing 
plate and the face of the crystal array be plane 



Figure 53. Lapping of a crystal array in 
preparation for mounting it on a porcelain- 
coated steel backing plate. 


of a square cross section immediately beneath 
the block of crystals and then reduces to a circu¬ 
lar cross section for the part which extends into 
the metal-air cell. A transducer employing such 
units has been illustrated in Figures 29 to 32 of 
Chapter 1. The advantage of Duralumin is pri¬ 
marily one of reducing the weight since it has 
an acoustic velocity approximating that of steel. 
The air space between the backing plate and the 
metal cap is sealed by means of cement and a 
thick rubber washer. The insulating layer be¬ 
tween the crystals and the metal backing plate 
consists of a thin sheet of bakelite impregnated 
fabric. The thickness is approximately in. 


Figure 55. A constructional view of a labor¬ 
atory pneumatic press as employed at UCDWR 
during the oven curing of an array bonded to a 
supporting structure. 

ciently flat was discussed in Section 8.7.2. The 
lapping of the surface of the crystal array 
which is to be cemented to the backing plate is 
illustrated in Figure 53. For a tool, a sheet of 



Figure 54. Spraying the initial 0.002-in. layer 
of bakelite cement on a porcelain backing plate 
and on a crystal array. 

to within a very few thousandths of an inch. 
The grinding of the surface of a porcelain back¬ 
ing plate in order to assure that it was suffl- 












PREPARATION OF ARRAYS 


323 


240-grit silicon carbide may be cemented to a 
piece of 1 / 2 -in. plate glass or held as shown in 
the illustration. Since this fine grade of abra¬ 
sive paper quickly becomes loaded with pow¬ 
dered ADP or RS, it should be cleaned fre¬ 
quently by means of a stream of compressed air. 
The lapping process should be continued until 
the face of the array is flat, as judged by a good 
quality surface plate. 

In the light of past experience at UCDWR, 
the following cementing technique for attach¬ 
ing RS crystals to porcelain-coated backing 
plates is suggested. Both the porcelain surface 
of the backing plate and the face of the crystal 
array are covered with a 0.002-in. layer of bake- 
lite or other appropriate cement. The detailed 
method of application of various cements has 
been discussed elsewhere but it may be pointed 
out here that a spray technique is to be pre¬ 
ferred as illustrated in Figure 54. This initial 
coat of bakelite cement is allowed to dry quite 
thoroughly by exposure to the air for a mini¬ 
mum of 60 min. Just previous to the final as¬ 
sembly of the crystal array on the plate, an ad¬ 
ditional thin layer of cement should be sprayed 
on the porcelain only and allowed to dry until 
tacky. The crystals should then be positioned 
on the plate and held firmly in place with a uni¬ 
form pressure of at least 25 psi. It may prove 
to be better to use a still higher pressure. This 
pressure is best applied by means of a pneu¬ 
matic press. A press found suitable for this ap¬ 
plication at UCDWR is illustrated in Figure 55. 
Uniform pressure on each crystal is assured by 
this type of construction in which a layer of 
rubber is covered with a layer of heavy canvas 
and cemented around the edges to the top plate 
of the press. In addition, both are held against 
the top plate by a metal frame. After thoroughly 
warming the assembly of bars, crystals, and 
clamps to 115 F in an oven of low humidity, it 
should be placed in an oven at 115 F and 60 
per cent relative humidity. The preliminary 
warming in a low humidity oven is essential in 
order to prevent condensation at the moment 
they are placed in the 60 per cent relative hu¬ 
midity oven. At the end of a period of at least 
12 hr (24 hr is customary), the assembly should 
be removed from the oven and allowed to cool 
down to room tem.perature before the pres¬ 


sure is released or the clamps are removed. 

For ADP assemblies, the curing temperature 
may be increased to 80 C without reducing the 
curing time. Before attempting either RS or 
ADP bonding to backing plates. Sections 8.6.2 
to 8.6.4 should be read. 

For examples of crystal arrays mounted on 
porcelain backing plates, reference may be 
made to Figures 33 and 34 of Chapter 1. The 
former shows ADP crystals both on bars and on 
a plate in the same transducer; the latter shows 
a smaller motor of the UCDWR-type GD class. 
Still another example is Figure 49 of the pres¬ 
ent chapter, already discussed in connection 
with lobe suppression. 


^ Other Supports 

Fronting Plates 

Owing to very meager experience with front¬ 
ing plates at UCDWR it is difficult to comment 
on the methods of assembling crystal arrays for 
this type of transducer. However, it would seem 
that no new problems are involved as far as the 
mode of assembly of the crystal array is con¬ 
cerned that have not been discussed already in 
preceding sections in connection with backing 
plates. 

The use of rubber windows as fronting plates 
will be discussed in the following section from 
the standpoint of inertia drive units. Plexiglas 
or Lucite has been used as a fronting plate in 
one or two instances. One such transducer is 
the Model 45-AX-l high-power projector de¬ 
signed by BTL in which a i/4-in. thick Plexiglas 
diaphragm was used as a fronting plate, appar¬ 
ently with the intention of imparting a broad¬ 
band characteristic to the radiation. 

Rubber (Inertia Drive) 

A number of inertia-drive transducers have 
been designed at UCDWR. One simple type, EP, 
produced in some quantity is illustrated in Fig¬ 
ure 56. This unit consists of a large number of 
thin ADP crystals which have been bonded to¬ 
gether in a solid block. Long strips of silver 
foil are cemented between adjacent rows of 
crystals to serve as electrodes. By having alter¬ 
nate strips protrude from opposite sides of the 




324 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


crystal array the wiring arrangement has been 
very much simplified as is clearly brought out in 
the illustration. The jig which clamped this 



Figure 56. A flat inertia drive ADP array, 
bonded directly to a rubber window by the Cycle- 
Weld process. Note polarity marks, also the long 
silver foil strips which serve both as electrodes 
and as a means of wiring the array. 

crystal assembly together while oven-curing the 
cement joints between the foils and the crystals 
is shown in Figure 37 of Chapter 1. 

The entire crystal assembly was bonded as a 
unit to a rubber diaphragm by means of the 


crystal motor in its case, either an air space 
was provided immediately behind the crystals 
or they were permitted to rest against a layer 
of Cell-tite rubber. For assembly drawing, see 
Figure 82. 

An interior view of a cylindrical inertia-drive 
transducer constructed on the same principle 
is photographed in Figure 40 of Chapter 1. In 
this case the crystals have been bonded to the 
interior of a reinforced rubber cylinder (see 
Figure 68) by means of the special jig illus¬ 
trated in Figure 51. A further discussion on 
the construction of this particular transducer is 
contained in Section 8.7.1. 

A third type of inertia-drive transducer de¬ 
signed at UCDWR is still in the trial stage. 
Since it possesses promising features from the 
standpoint of transducer construction, it will 
be briefly described. The crystals, either indi¬ 
vidually or grouped into strips, are bonded to a 
thin sheet of rubber by the Cycle-Weld process. 
One stage of the assembly process for an array 
consisting of individually foiled crystals is 
shown in Figure 57. After the array has been 
bonded to the rubber it may readily be formed 
into a circular contour as illustrated in Figure 
58. In order to obtain inertia-drive character¬ 
istics, a sheet of Cell-tite rubber is placed be- 




Figure 57. One stage in the construction of a 
cylindrical array. Assembling individual crystals 
in a jig prior to bonding them to a thin rubber 
mounting strip by the Cycle-Weld process. See 
Figure 58. 


Figure 58. Forming the completed array of 
Figure 57 into a circular configuration. The addi¬ 
tion of a layer of Cell-tite rubber beneath the 
solid rubber mounting strip results in an 
essentially inertia driven transducer. 


Cycle-Weld process described in Section 8.6.6. 
A part of the rim and diaphragm has been cut 
away in Figure 56 in order to show the manner 
in which the rubber diaphragm has been molded 
into the steel rim. The thickness of the neo¬ 
prene diaphragm was in. In mounting this 


neath the thin mounting strip of rubber to 
which the crystals are bonded. Another illus¬ 
tration which shows a completed transducer 
constructed on this principle occurs in Figure 
62. After the strip containing the crystals had 
been coiled into a circular configuration and 






PREPARATION OF ARRAYS 


325 


cemented to a central metal core, additional 
support was given to the array by wrapping 
nylon thread about its circumference. The non¬ 
radiating edges of all crystals are blanked with 
Cell-tite rubber. 

Stacks 

A common type of stack transducer which has 
been built in large quantities at UCDWR is 



Figure 59. A typical stack crystal array as 
developed at UCDWR. 


illustrated in Figure 59. The essential features 
have already been discussed in Section 8.7.1. 
For the most part stack-type arrays have been 
mounted within layers of Corprene and inserted 
in tin can cases of the kind discussed in Section 
8.8.3. The stack unit is held securely in position 
in its cylindrical case by cementing to the top 
and bottom of the crystal-stack disks of rubber, 
plastic, or Corprene which have the same in¬ 
ternal diameter as the can. 

Blanking of the nonradiating faces of the 
crystals has been done usually by means of Cor¬ 
prene or Cell-tite rubber. The use of Corprene 
has been more common at UCDWR since it is 
somewhat more convenient to handle. By using 
narrow strips of Corprene along each edge of 
the crystal stack as shown in Figure 59, one re¬ 
duces the area of contact between the Corprene 


and the vibrating crystals. The wiring arrange¬ 
ment should be sufficiently clear from the figure. 
In similar stack units constructed at the Brush 
Development Company it has been the practice 
to have the sheets of Corprene lying flat against 



Figure 60. The UCDWR type 24C1Y1 (form¬ 
erly CCZIO) stack transducer of Y-cut Rochelle 
salt crystals. 

the nonradiating faces. Narrow slots cut in the 
Corprene sheets permit the foil ends to be 
brought out for soldering lugs. The tin-foil sol¬ 
dering tabs extend through the slots and are 
folded down and pressed against the outside of 
















326 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


the strip before the conducting wire is soldered 
to them. 

Another example of a stacked array is shown 
in Figure 60. In this case the foil tabs are 
brought out along the radiating face, but the 
wire is so small that it does not interfere with 
the radiation field. Contact of the curved Cor- 
prene strip with the crystals has been avoided 
everywhere except at the central, relatively im¬ 
mobile, portion of the crystals by the use of a 
square Corprene rail. A Lucite plate having the 
same dimensions as one of the individual crys¬ 
tals is used at the bottom of the crystal stack 
and a similar Lucite plate except somewhat 
longer is used on top of the crystal stack. The 
additional length in the Lucite plate at the top 
has permitted holes to be drilled in it which act 
as supports for the lead wires. In addition to 
protecting the crystals on either end of the 
stack, these Lucite plates also serve to keep the 
narrow silver foil in contact with the end crys¬ 
tals. The narrow silver foils are cemented 
lightly between each pair of crystals, thus in¬ 
creasing the mechanical strength of the stack 
as well as anchoring the foils. An attempt to 
replace the tin-foil electrodes and the narrow 
foil strips in this transducer by a single silver 
foil between each pair of crystals proved un¬ 
successful in that it caused a lowering of the 
transducer output by several db. However, in 
this attempt all of the crystals in the stack were 
securely cemented together into a solid block. 
Further investigations along this line must be 
conducted before a final conclusion can be 
reached. 

® " ^ Wiring of Arrays 

Choice of Materials 

The principal factor governing the choice of 
wiring materials is resistance to corrosion in 
the presence of castor oil which is also in con¬ 
tact with rubber and RS or ADP crystals. At¬ 
tention was focused on this problem in the early 
days of transducer design because it was ob¬ 
served that copper wire corroded in RS trans¬ 
ducers. Whether this corrosion was due to in¬ 
teraction with the castor oil itself or to the addi¬ 
tional presence of RS and/or rubber containing 


sulfur is not known definitely to the writer. 
However, there is a well-established tradition 
that plain copper wire should not be used in 
transducers containing castor oil. Accordingly, 
it has been the custom to use well-tinned copper 
for this application. Silver wire has been used 
for wiring crystal assemblies at the UCDWR 
laboratory, apparently without any evidence of 
corrosive action. 

Where the individual crystal electrodes are 
not furnished with tabs to permit soldering, it 
has been customary to connect the electrodes of 
the individual crystals with narrow strips of 
metal foil. The material used for these strips 
differs from one laboratory to another. Current 
practice at NRL is to use 0.002-in. nickel foil, 
at BTL 0.0007-in. gold-plated German silver 
which is given a ripple finish to improve the 
contact with the crystal electrode, at the Brush 
Development Company 0.001-in. gold-plated or 
tin-plated silver foil, and at UCDWR 0.0017-in. 
pure silver foil. Since all of these materials 
have been employed satisfactorily in existing 
equipment, any choice would seem to be a mat¬ 
ter of individual preference. However, the qual¬ 
ity of the electric contact is the most impor¬ 
tant consideration. In this respect, long experi¬ 
ence at BTL has shown that gold forms the best 
low-resistance contact and the one least subject 
to corrosion. For high-power applications, the 
gold contacts may even prove essential. 

Electric Contact Strips 

The manner in which the electrodes of the in¬ 
dividual crystals are connected by long strips of 
foil has already been discussed in Section 8.7.1 
for one type of plane crystal array and relevant 
illustrations occur in Figures 47 and 61. Simi¬ 
larly, for stack-type arrays. Figure 59 will show 
clearly how silver strips which make contact 
with the tin-plated electrodes are brought out to 
permit soldering to the wire leads. Some other 
types of arrays may be quickly and easily built 
up in a similar manner. 

Good electric contact to the electrode has 
been assured at UCDWR by imparting a sand¬ 
paper finish to the silver foils. This has been 
accomplished by laying the strips of silver foil 
on a piece of No. 2 emery cloth, placing a piece 
of Corprene approximately % in. thick on the 




PREPARATION OF ARRAYS 


327 


foil, and pressing this assembly in a book press. 
An imprint of the many small protrusions on 
the emery paper is left in the foil. These sharp 
projections materially reduce the contact re¬ 
sistance between the silver and the tin-foiled 
surface of the crystal. Other laboratories have 
treated contacting foils in an analogous man- 




Figure 61. Two stages in the wiring of a crys¬ 
tal array. Above: Threading a wire through 
holes in the foil strips. Below: Soldering the 
foils to the wire. A minimum of solder should be 
used with a minimum of heat in order to avoid 
fracturing the crystals. 

ner, using the materials mentioned in Section 

8.7.4. 

It is usual to cement these contacting foils to 
the electrodes in order to make their position 
secure. However, the cement layer should be 
extremely thin. 

Soldering Precautions 

For soldering connections inside transducers 
a good grade of soft solder, preferably with the 
eutectic 63-37 composition, is recommended. 
But 60-40 or even 50-50 have been found satis¬ 
factory. Either rosin-core solder or a rosin- 


alcohol flux should be employed. It is obvious 
that all solder connections should be done in 
such a fashion as to guarantee a permanent 
joint. 

Since both RS and ADP crystals are quite 
readily fractured if subjected to large tempera¬ 
ture gradients, soldering operations must be 
carried out carefully to prevent fracturing the 
crystals. This is particularly true for silver foils 
where the connecting wires are soldered di¬ 
rectly to tabs on each individual crystal. The 
minimum of heat and the minimum of solder 
consistent with a reliable electrical connection 
should be the rule. It must be kept in mind, how¬ 
ever, that transducers are subjected to mechani¬ 
cal vibration and also to depth charges; hence, 
the soldered connections must be mechanically 
rugged. 

Wiring Arrangements 

In arranging the wiring for a crystal array, 
attention must be given to the questions of volt¬ 
age insulation and wiring capacitance. In oil- 
filled transducers, voltage insulation does not 
usually constitute a major problem; in air-filled 
units which operate as a source and therefore 
at relatively high voltage, the problem may be 
important. The seriousness of the voltage in¬ 
sulation question depends also on the type of 
transducer design. For example, in a 2 to 1 lobe- 
suppressed array, illustrated in Figure 50, there 
is an intrinsic insulation difficulty which has 
had to be met by a wider spacing of the crystal 
groups between the two zones. This spacing in 
the array of Figure 50 amounts to % in. For 
this reason, the 3 to 1 scheme of lobe suppres¬ 
sion has been favored at UCDWR (see Figure 
49, also Figure 33, of Chapter 1). 

In most transducers the effect of the wiring 
capacitance on acoustic performance is prob¬ 
ably negligible. In some small units, however, 
the wiring capacitance may play an important 
role. Since special cases of this kind must be 
studied individually, there is nothing particu¬ 
larly helpful which can be added on this topic 
in this section. 

Since important technical considerations are 
frequently not involved, the arrangement of 
wiring in transducers is often dictated on the 
basis of simplicity arguments. Where space per- 

































328 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


mits the wiring may be arranged between rows 
or banks of crystals as in the lobe suppressed 
crystal array illustrated in Figure 50. Usually 
it is convenient to have the wiring near the edge 
of the crystal array as illustrated in Figure 56 
or in Figure 33 of Chapter 1. In still other 
transducers it may be preferable to have the 
wires directly over the radiating face of the 
crystal array as in Figure 49 and Figure 60. In¬ 
asmuch as the diameter of the wires generally 
constitute a very small fraction of the wave- 


This, however, does not constitute a practical 
solution to the problem for naval equipment op¬ 
erating at sea. The next best procedure would 
be to use a gas, preferably air, at the same pres¬ 
sure as its surroundings. This type of construc¬ 
tion has been used in many different designs of 
transducers where air filling is permissible. In 
general, it is more difficult to secure a satisfac¬ 
tory watertight seal with air-filled transducers 
than with liquid-filled types, especially where 
the former operate at great depths in the water 



Figure 62. Soldering wires to the foil strips in a UCDWR cylindrical transducer. Note the use of metal- 
glass terminal seals; also the grooves provided for 0-ring hydraulic gaskets with which to seal the com¬ 
pleted assembly into its cylindrical housing. 


length of the radiation, there does not seem to 
be any valid technical objection to this pro¬ 
cedure, especially where bare wire is employed. 
The use of insulating material which might act 
as a pressure release immediately in front of the 
radiating face would be objectionable, of course. 

Acoustic-Isolation Materials 
Free Gas 

The highest degree of acoustic isolation ob¬ 
tainable within an array consists in having the 
individual crystal elements, except the radiat¬ 
ing face, surrounded by an evacuated space. 


and therefore are subjected to a large pressure 
differential. Attempts to equalize the pressure 
by various contrivances, while not unsuccessful, 
have usually led to complicated and awkward 
devices. 

Gases other than air may prove desirable in 
special cases. Where voltage breakdown is a 
problem, for example, recourse may be had to 
Freon (dichloro-difluoro methane). Freon will 
withstand approximately three times as high a 
voltage as air at atmospheric pressure. While 
Freon has been tried in a few experimental 
transducers at UCDWR, its performance in the 
field remains unknown. 









PREPARATION OF ARRAYS 


329 


Metal-Air Cells 

In liquid-filled transducers it is not feasible 
to surround the nonradiating faces of each crys¬ 
tal with air, but provision for acoustically iso¬ 
lating the ends opposite the radiating face may 
be met in any one of several ways. One rela¬ 
tively simple procedure is to provide an air 
layer sandwiched between two metal plates. 
These metal plates may be either a few thou¬ 
sandths of an inch thick and sealed by soldering 
around the edges or they may be sufficiently 
thick, perhaps V 2 in., to withstand relatively 
high pressures. Where thin metal walls are 
used it may be necessary to provide internal 
inserts which will prevent their collapse under 
conditions of high pressure. If these metal-air 
sandwiches must be subjected to a vacuum dur¬ 
ing the liquid-filling process of the transducer, 
it will probably be necessary to weld the inserts 
to prevent the walls from expanding. 

A unique type of cellular construction in 
which an air layer is maintained in connection 
with a backing plate is shown in Figure 52. Con¬ 
structional details of this backing plate unit 
have been discussed in Section 8.7.2. 

Cellular Rubber 

Much of the advantage of an air layer for 
acoustic isolation may be obtained by the use of 
cellular rubber. That used at UCDWR had the 
trade name Cell-tite. In this material the air is 



Figure 63. A Cell-tite rubber block completely 
sealed in a molded rubber sheath in order to pre¬ 
vent the diffusion of air from its cellular matrix. 
(Bell Telephone Laboratories.) 

contained in small noncommunicating bubbles 
or cells incorporated into a rubber matrix. The 


material may be readily cut with shears or a 
razor blade and is available in sheets a yard 
wide and in thicknesses ranging from in. to 
1/4 in. or more. 

Two questions arise in connection with the 
use of cellular rubber in liquid-filled trans¬ 
ducers. One involves its possible interaction 
with the liquid and the other involves the ques¬ 
tion of gaseous diffusion through the thin walls 
of the individual cells. The latter problem de¬ 
pends in part for its answer on the individual 
application. For transducers operating at great 
depths it is clear that the gas would be sub¬ 
jected to high hydrostatic pressures and it 
would seem only a question of tim.e before an 
appreciable part of the gas could be lost by dif¬ 
fusion. In addition, one can envisage a loss of 
efficiency at great depths should the cellular 
rubber become unduly compressed. 

Attempts have been made to reduce the likeli¬ 
hood of the gas escaping from the cellular struc¬ 
tures by coating the rubber surface with a more 
impervious substance or by adding a thicker 
layer of rubber on the outside of the cellular 
matrix. In Figure 63 is shown a cross section of 
a sample resulting from one such attempt by 
BTL in which the cellular rubber has had an 
approximately layer of solid rubber 

molded completely about it. Although this is a 
step in the right direction, efforts must still be 
exerted to find thinner layers of satisfactory 
cellular structures. To pursue this matter fur¬ 
ther it would be well to consider molding about 
the cellular matrix other materials which are 
even more impervious to gaseous diffusion. In 
particular, some of the synthetic materials such 
as butyl rubber or Koroseal should be tried. 

In a great number of transducers designed 
at UCDWR it has been specified that strips of 
Cell-tite rubber be placed on all of the non¬ 
radiating edges of the crystals with the excep¬ 
tion of the electrode faces. In order to hold the 
Cell-tite rubber in place, it has frequently been 
necessary to bond them to the crystals with 
some type of cement. For the most part bake- 
lite cement has served this purpose. In attach¬ 
ing Cell-tite rubber on the narrow crystal sur¬ 
faces between the electrode faces, particular 
attention must be paid to preserving high val¬ 
ues for the electrical resistivity. 




330 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


This electrical resistivity may be adversely 
affected in several ways. 

1. The surface coating on the Cell-tite rubber 
may be contaminated with conducting ma¬ 
terials. At UCDWR it has been found neces¬ 
sary to thoroughly wash and scrub the surface 
of the rubber with cheesecloth containing an 
organic solvent such as benzene. 

2. It is important to avoid fingerprints, both 
on the surface of the rubber as well as on the 
crystals. 

3. Any moisture which may be trapped 
beneath the surface of the crystal and either 
the layer of cement or the rubber may seriously 
increase the surface leakage. 

This last point has been discussed in detail 
in Section 8.2.4, including the graph of Figure 
3. The curves on this graph applied to an RS 
crystal which had been coated with Vulcalock 
cement and are not necessarily applicable to 
ADP. However, it has been found in voltage¬ 
testing ADP crystals that arcing is more likely 
to occur through the cement layer than across 
the somewhat greater air path around the in¬ 
sulating material. It would appear preferable 
if possible to avoid cement altogether in con¬ 
nection with Cell-tite rubber. This can be done 
in some transducers which have a proper spac¬ 
ing of the crystals merely by wedging the 
individual strips of Cell-tite rubber between 
them. There is little tendency for these strips 
to become dislodged in an oil-filled transducer. 

In bonding cellular rubber to large surfaces, 
it should not be stretched. Otherwise it may pull 
away later and not cover the desired area com¬ 
pletely. When heated under pressure, cellular 
rubber collapses. It is added to crystal arrays 
following the oven-curing processes. 

Cork and Cork-Rubber Compositions 

The use of cork naturally suggests itself for 
purposes of acoustic isolation because it con¬ 
tains a large percentage of air. Although 
natural cork has been used to some extent for 
this application, a number of cork-rubber com¬ 
positions are available which are much superior. 
This superiority arises from selecting a matrix 
material which is better than natural cork as 
regards its imperviousness to both gases and 
liquids. At the same time it is important to 


select a cork-rubber composition which is re¬ 
sistant to the liquid involved and which 
possesses a high voltage-breakdown value. A 
commercial product which has been used to 
advantage at several laboratories has a 
neoprene-cork composition (Armstrong type 
DC-100). A sample in- thick has been 
known to withstand a breakdown test at 30 kc 
of 10,000 V rms. 

These cork-rubber compositions are available 
in large sheets in thicknesses from to 1 / 4 , in. 
or more. They may be cut to any desired size 
very conveniently. No data are available as to 
whether any particular composition excels 
acoustically. It was felt by the designer at one 
laboratory that the best composition to use 
should have a Shore durometer test between 
50 and 60. 

The remarks in Section 8.7.5 with reference 
to the cementing of Cell-tite rubber to crystals 
are also applicable to cork and cork-rubber 
compositions. 


” ‘ ® Inspection and Test of Arrays 
Visual Inspection 

A careful visual inspection of completed 
arrays may result in the detection of faulty con¬ 
struction of several different kinds. One of the 
more important observations that can be made 
is an examination of the quality of the cemented 
bond. To facilitate inspection it is frequently 
advantageous to place a thin oil film on the 
radiating face of each crystal in order to obtain 
a clearer view- of the bond. Where an appre¬ 
ciable number of the crystals are found to be 
improperly bonded, the entire array must be 
rebuilt. If only 1 or 2 per cent of the crystals 
in an array are improperly bonded, the unit 
would probably be considered acceptable. In 
some cases a few faulty crystals in an assembly 
can be replaced and rebonded satisfactorily 
without dismantling the whole unit, but this is 
usually difficult. 

A thorough inspection should be made to see 
that all the electrical connections are secure 
and that there is an absence of solder or other 
loose dirt particles. Where polarity markings 
are such as to be visible, each crystal should be 



PREPARATION OF ARRAYS 


331 


checked to see that its polarity is correct. All 
isolating strips of Corprene or Cell-tite rubber 
specified in the design drawings should be 
checked for proper location. 

The completed array must be thoroughly 
cleaned before it is permitted to pass inspection. 
Especially, excess cement and finger prints are 
likely to be present. It is not sufficient that 
crystals merely look clean, since conducting 
films or filaments may be quite invisible. If 
organic solvents are used to clean the crystals, 
proper regard must be paid to the solubility of 
the cemented bonds and other assembly com¬ 
ponents. This usually means that only a cloth 
dampened with solvent is employed for wiping 
the crystal surfaces, not an immersion of the 
entire assembly. The final check on the cleanli¬ 
ness of a crystalline array is the electrical test¬ 
ing for d-c resistance and voltage breakdown 
discussed in Section 8.7.6. 

Polarity of Crystals 

Since the polarity of a crystal determines the 
phase relationship between electrical impulse 
and mechanical action, the polarity of each 
crystal is determined prior to its assembly into 



Figure 64. Inspecting the individual crystals on 
a backing bar for correct orientation by means 
of a polarity indicator, 

an array as already discussed in Section 8.3.9. 
During the construction of large arrays, in¬ 
volving perhaps hundreds of crystals, numerous 
chances for errors arise. Apart from possible 
mistakes in the original polarity indications on 
each crystal, the crystals may be installed in an 
assembly in a reversed position or the subse¬ 


quent wiring may be incorrect. It is therefore 
highly desirable to recheck the polarity of each 
crystal in its permanent position in the com¬ 
pleted assembly. 

This recheck of the polarity of each individual 
crystal may be accomplished in the manner 
shown in Figure 64. The indicating equipment 
for this purpose has already been described in 
Section 8.4.8. Each crystal is given a sudden 
push on its radiating face by means of a rubber- 
tipped pencil and the polarity indication is 
noted on the meter. In very large arrays, which 
may involve 100 to 500 or more crystals in 
parallel, the increased capacitance of the elec¬ 
trical circuit will tend to obscure somewhat the 
polarity indication which results from the 
voltage generated by the single crystal under 
test. If positive indications cannot be obtained 
when all the crystals of the complete array are 
in parallel, individual rows of crystals should 
be disconnected from the circuit and tested. 
Such an individual row of crystals under test is 
illustrated in Figure 64. 

It should be pointed out that false indications 
of polarity may occur occasionally if the indi¬ 
vidual crystals under test are bonded to other 
crystals in a solid array. In assemblies where 
the crystals have been tightly packed, cases 
have been noted where the deflection of the 
polarity indicator made an individual crystal 
appear to be reversed in position. Removal and 
independent test of such a crystal has proved 
oftentimes that it had been correctly polarized, 
and also properly installed and wired in posi¬ 
tion. In cases of this kind, it must be assumed 
that pressure exerted on the end of the tested 
crystal resulted in distortion of its neighboring 
crystals to the extent that their out-of-phase 
output exceeded that of the crystal under test. 

The correctness of the polarity of each crystal 
in a group can be determined also by using a 
probe microphone. The entire array of crystals 
is driven by an oscillator and the microphone 
probe is placed successively on the radiating 
face of each crystal of the array. A description 
of the probe-microphone equipment and the 
technique involved in its use occurs in detail in 
Section 9.2. Briefly, however, the probe micro¬ 
phone consists of two tiny piezoelectric crystals 
mounted in a small holder so that any mechan- 








332 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


ical pressure exerted on them can be amplified 
and read on electric meters which indicate both 
magnitude and phase. 

At UCDWR the probe has been coupled me¬ 
chanically to the radiating crystal surface by 
a thin film of castor oil. An oscilloscope coupled 
both to the probe and to the driving signal of 
the oscillator has been used to indicate phase 
relationship and thus to determine whether the 
individual crystals of the array are radiating 
in phase. 

D-C Resistance 

The d-c resistivity of RS and ADP has been 
discussed in some detail in Sections 8.2.4, 8.2.8, 
and 8.5.7. It was seen that the d-c resistance 
varied with the temperature and humidity, 
particularly in the case of RS. In completed 
arrays consisting of a few hundred crystals the 
d-c resistance would be expected to depend in 
a known way on the circuit design and on both 
the dimensions and the number of crystals. 
Consequently a fair approximation to the d-c 
resistance expected for a particular transducer 
should be calculable from a knowledge of the 
kind of crystals, their dimensions and number, 
and their electrical connections. In practice, 
however, it will be found that the d-c resistance 
of a crystal array will vary markedly with the 
quality of the technique used in its assembly. 
For example, a large array of X-cut RS crystals 
may have a resistance as low as 20 megohms 
when constructed in a casual manner without 
special precautions. An identical crystal array, 
but constructed with care, may possess a d-c 
resistance as high as 1,000 megohms. To attain 
the higher value, it is especially important to 
refrain from touching the interelectrode sur¬ 
faces of the crystals with the bare hands and 
to eliminate adsorbed moisture. 

The presence of water vapor is a most im¬ 
portant factor with RS. For instance, newly 
constructed arrays of RS crystals may have a 
very low resistance, perhaps only 100,000 or 
200,000 ohms. When they are subjected to a 
vacuum for a time their resistance will gradu¬ 
ally increase until it reaches a maximum value 
of perhaps 2,000 megohms or better. Upon 
being removed from the vacuum chamber this 
resistance value will drop somewhat but prob¬ 


ably not below 100 megohms. In general, it is 
considered satisfactory if large RS transducers 
have a d-c resistance of 50 megohms or more 
when first constructed, although this value will 
be expected to increase upon being subjected to 
a vacuum. ADP crystals, however, are not so 
sensitive to relative humidity conditions so 
that little if any improvement can be expected 
in the d-c resistance of ADP arrays upon plac¬ 
ing them in a vacuum chamber. 

In addition to measuring the d-c resistance 
between the terminals of a crystal array, it is 
also desirable to check the resistance from each 
terminal to ground. 

Resistance readings at UCDWR have been 
made usually with vacuum-tube ohmmeters. In 
general, precision is not an important con¬ 
sideration. 

Discussion of the d-c resistance measure¬ 
ments to be made on transducers in their com¬ 
pleted state will be discussed in Section 8.9.5. 

Capacitance 

Capacitance values at 1,000 c (Cj,) for indi¬ 
vidual crystals of RS and ADP were discussed 
in Section 8.5.4. From the information given it 
is possible to calculate the capacitance to be 
expected of an assembled array from the kind, 
number, and dimensions of the individual 
crystals and the manner in which they have 
been connected electrically. This calculated 
value may be checked with experimentally 
determined values for the completed crystal 
array. This type of check has not customarily 
been made at UCDWR except occasionally on 
transducers which have been finally cased and 
oil-filled. 

Admittance 

Admittance measurements on single crystals 
have been discussed in Section 8.5.5. The elec¬ 
tric circuit for the determination of admittance 
characteristics was also given, together with a 
sample admittance curve for a single ADP 
crystal. It is true that admittance curves have 
greater significance in connection with single 
crystals than when they are determined for an 
assembly of many crystals bonded to a support¬ 
ing structure and perhaps to each other. In some 
cases, nevertheless, a rough indication of the 



HOUSINGS AND ACCESSORIES 


333 


efficiency of a crystal transducer can be 
obtained from an admittance curve, particularly 
if the efficiency is high. For a highly efficient 
transducer, a rise of 2 or 3 db in the admittance 
curve may occur at resonance. However, if no 
increase in slope of the admittance curve can 
be discerned at resonance, the transducer might 
still possess an efficiency of 50 per cent. 

High Voltage 

High-voltage specifications for individual 
crystals were discussed in Section 8.5.8. For a 
completed array, it seems to be too much to ex¬ 
pect that the breakdown voltage of an entire 
assembly should be as high as the minimum 
breakdown voltage of the individual crystals 
which comprise it. In a number of experimental 
transducers constructed at UCDWR, the indi¬ 
vidual crystals used were tested at a voltage 
higher than that at which the completed trans¬ 
ducer was to be operated, but near the expected 
breakdown voltage of individual crystals. It was 
found that in almost all cases the assembled 
arrays failed at voltages considerably below the 
test voltage for the single crystals. The explana¬ 
tion for this behavior probably lies in the con¬ 
tamination of some crystal surfaces during the 
course of construction, but, generally speaking, 
voltage breakdown is somewhat unpredictable. 

Attempts have been made at UCDWR to 
determine maximum safe operating voltages 
for the three common piezoelectric crystals. In¬ 
dividual ADP crystals i/i in. thick in oil were 
found to withstand 20,000 v rms in almost every 
case. Many of them tested higher than 30 kv 
and a few did not break down even at 40 kv. 
When failure occurred the breakdown usually 
took place through the body of the crystal. 
When the individual crystals are built into an 
array it might be expected that the array would 
withstand a test voltage of 20 kv. Nevertheless, 
this is not the case and, in general, it has been 
found for ADP crystals in. thick that approx¬ 
imately 5,000 V rms constitutes an upper limit 
for a safe operating voltage unless special pre¬ 
cautions are taken during construction. 

Unless special demands are to be placed upon 
ADP transducers, it would seem that voltage 
tests well in excess of the operating range 
should be made on the crystal array, provided 


this test voltage does not exceed the specifica¬ 
tions given in Section 8.5.8. 

Single Y-cut RS crystals in. thick will 
normally withstand a test voltage of 20 kv rms. 
In practice, however, the maximum safe operat¬ 
ing voltage for RS transducers has been in the 
neighborhood of 2,000 v. In several experi¬ 
mental transducers constructed at UCDWR, it 
was hoped that much higher operating voltages 
could be used as a consequence of the careful 
technique employed in their construction. As a 
matter of fact, breakdown voltages in excess of 
6,000 V rms were obtained in one or two cases. 
As the result of much experience with RS trans¬ 
ducers it has been found that the maximum safe 
operating voltage of 2,000 v rms is about as 
much as can be expected for i/4-in. crystals. 
Crystals of other thicknesses would withstand 
proportional operating voltages. 

It may be inconvenient to make an inspection 
test of assembled crystal arrays much above 
their rated operating voltage in that it will 
ordinarily be necessary to immerse them in a 
liquid. The use of most organic solvents for the 
immersion liquid, such as carbon tetrachloride 
mentioned in the specifications for testing in¬ 
dividual crystals, is not permitted owing to 
their deleterious effect on most cement joints. 
One procedure would be to use the regular 
transducer liquid for this purpose, but this is 
inconvenient where further work on the array 
is contemplated. One possibility that suggests 
itself, if the test voltage is not appreciably 
higher than the air breakdown voltage for the 
spacings involved, is the use of a Freon atmos¬ 
phere. The breakdown voltage of Freon at 
atmospheric pressure is approximately three 
times that of air. 

« « HOUSINGS AND ACCESSORIES 
^ ^ Specifications and Tests 

Several factors must be considered in the 
choice of a material for housing transducers. In 
general, the acoustic properties of the material 
selected are unimportant, except for the window 
through which the radiation enters the water. 
When crosstalk transmitted through the case 
proves to be a problem, as it has in some com- 




334 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


plicated transducers which contain two or more 
transmitting and/or receiving units in the same 
housing, the acoustic properties of the case ma¬ 
terials may have to be taken into consideration. 

One of the most important requirements for 
transducer cases is that they must be free from 
leaks. It seems to be quite difficult to fabricate 
cases in such a manner that they can be guar¬ 
anteed leakproof; only strict attention to detail 
during the course of manufacture will result in 
satisfactory performance. The quality of metal 
castings with respect to leaks will be discussed 
in Section 8.8.2. All transducer housings should 
be thoroughly tested for leaks before using. 
This is probably done most conveniently by 
using 60 to 70 psi air pressure inside the case 
and inspecting the outside for bubbles, either 
while immersed in water or while the exterior 
is wet with soap solution. Although the housing 
may be subjected to considerably higher pres¬ 
sures in actual use, it is too hazardous to test 
at a still higher pressure unless special pre¬ 
cautions are taken to safeguard personnel. 

Adequate mechanical strength in a trans¬ 
ducer housing is a matter of design, but tests 
should be conducted to see that design specifica¬ 
tions are met. Partial tests may be made by 
filling the cases with liquid at the required pres¬ 
sure. Usually the specifications as to strength 
will be such that the test equipment available 
in the ordinary laboratory may be inadequate. 
Actual tests in the field will then be required. 
As an illustration of the factors encountered, 
mention may be made of transducer cases in¬ 
tended for rocket launching or which are 
launched from high-speed aircraft so that they 
strike the water with tremendous velocity. It is 
clear that these conditions would be difficult to 
duplicate in the laboratory. Even in the case of 
ship-mounted transducers, which are subjected 
to rough seas and perhaps to depth charges, it 
will probably also be desirable to conduct tests 
under operating conditions. 

Corrosion resistance will always be an im¬ 
portant factor in the selection of materials for 
underwater operation. This problem may be 
attacked either by selecting a material which is 
least subject to corrosion, or by covering the 
transducer case with a corrosion resisting coat¬ 
ing. The adoption of the latter practice, if satis¬ 


factory, materially lessens the demands placed 
on the actual material of the housing. Corrosion 
resisting coatings and antifouling paints will 
be discussed in Section 8.8.5. Among workable 
metallic materials, best corrosion resistance at 
the present time is apparently found in some of 
the stainless steels and in the nickel-copper 
alloys, such as Monel and Inconel. Alloys con¬ 
taining more than 60 per cent copper are not 
likely to become fouled with marine organ¬ 
isms.^*^ The information in Table 2, which lists 
numerous elements and their alloys in a gal¬ 
vanic series for sea water, is reproduced from 
articles by LaQue^® and Cox.^"' Further dis¬ 
cussion and references on this important 
problem may be found in their articles. The 
importance of passive surface films on certain 
alloys is clearly demonstrated in this table. 


Table 2. Galvanic series for sea water. 


Magnesium 

Lead 

Magnesium alloj's 

Tin 

Zinc 

Muntz metal 

Galvanized steel or gal¬ 

Manganese bronze 

vanized wrought iron 

Naval brass 

Aluminum 52SH 

Nickel (active) 

Aluminum 4S 

Inconel (active) 

Aluminum 3S 

Yellow brass 

Aluminum 2S 

Admiralty brass 

Aluminum 53ST 

Aluminum bronze 

Alclad 

Red brass 

Cadmium 

Copper 

Aluminum A17ST 

Silicon bronze 

Aluminum 17ST 

Ambrac 

Aluminum 24ST 

70:30 copper nickel 

Mild steel 

Comp. G bronze (88% 

Wrought iron 

Cu, 10% Sn, 2% Zn) 

Cast ii’on 

Comp. M. bronze (88% 

Ni-resist 

Cu, 6.5% Sn, 4% Zn. 

13 per cent chromium 

1.5% Pb) 

stainless steel, type- 

Nickel (passive) 

410 (active) 

Inconel (passive) 

50-50 lead-tin solder 

Monel 

18-8 stainless steel, type- 

18-8 stainless steel, type- 

304 (active) 

304 (passive) 

18-8-3 stainless steel, 

18-8-3 stainless steel, 

type-316 (active) 

type-316 (passive) 


A streamlined transducer case which has 
been spun from sheet Inconel is photographed 
in Figure 65. The thick rubber window has 
been bonded inside a stainless-steel cylinder and 
has been given the contour of the Inconel hous¬ 
ing. The oc rubber was Compound M-163, whose 










HOUSINGS AND ACCESSORIES 


335 


composition occurs in Section 8.8.4. This trans¬ 
ducer case is made to oscillate by a mechanism 
enclosed in the remainder of the gear. 



Figure 65. A streamlined oscillating- transducer 
housing of Inconel developed at UCDWR. The 
ADP crystals have been bonded to the thick qc- 
rubber window, whose cylindrical exterior sur¬ 
face is clearly shown in the photograph. 

® ® “ Metal Castings 

Metal castings have been very widely used 
for transducer housings, partly for economical 
reasons, but also for the simplification of design 
which they permit. In general, castings would 
appear to be the best solution to the transducer 
housing problem except for their frequently 
poor quality with respect to porosity. Difficulty 
has been experienced in obtaining metal cast¬ 
ings entirely satisfactory in this respect. Ac¬ 
cordingly, all castings must be pressure tested 
in order to insure their freedom from leaks. 
Leaks may be detected usually by subjecting 
the castings to an air pressure of 70 psi while 
immersed in water or with the outside of the 
casting covered with a soap solution. This test¬ 
ing should be done immediately upon receipt of 
the castings, before any small cracks or holes 
have had a chance to become temporarily closed 
by oxide formation. Castings which contain 
small leaks should be discarded before any ma¬ 
chining time has been wasted on them. Since 
most crystal motors involve an outlay of a few 


hundred dollars, it is poor economy to try to 
salvage defective metal castings by attempts 
to make them waterproof. Only castings which 
are leak-free before and after machining by 
virtue of their own homogeneity should be used. 

Iron alloys have found the widest application 
in castings. Meehanite, a patented alloy, is 
especially dense in structure and has desirable 
machining qualities as well. It has been used 
with considerable success in transducers. 
Numerous examples of iron castings will be 
found in photographs of transducer cases 
throughout this volume. One common type at 
UCDWR is shown in Figure 79. 

Castings of stainless steel may prove an an¬ 
swer to this problem in the future, but they 
have seen very limited use so far. A photo¬ 
graphic illustration of a stainless-steel casting 
occurs in Figure 21 of Chapter 1. 

Castings of other metals have also been used 
for transducers, especially brass and bronze. 
These castings may be porous also and hence 
subject to leaks, particularly if the walls are 
thin. While it would seem simple to repair these 
leaks in the case of brass and bronze, again it 
seems undesirable to do so from an economic 
viewpoint. Exceptions to this conclusion might 
arise in the future, should better methods be¬ 
come available for the high-pressure impregna¬ 
tion of castings with plastic cements or other 
suitable substances. 

Cast aluminum corrodes very rapidly in sea 
water unless coated with a satisfactory protec¬ 
tive covering. The same is also true of pure 
aluminum and of many aluminum alloys. How¬ 
ever, all these materials have found extensive 
use for expendable applications, where they 
need last but a few hours. For the position of 
aluminum and aluminum alloys in the galvanic 
series for sea water, refer to Table 2 which 
appears in Section 8.8.1. 

« Tin-Can Cases 

Ordinary tin cans have been used extensively 
for housing expendable transducers. During the 
period 1941 to 1945 nearly 10,000 such trans¬ 
ducers have been manufactured and used satis¬ 
factorily. These cans are subject to corrosion, if 
exposed to sea water an appreciable length of 




336 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


time, but they are entirely acceptable for 
periods of a few hours, or, in the case of inter¬ 
mittent exposure, perhaps a few days. The wall 
thickness of the tin-plated steel used in these 
cans is approximately 0.010 in., so that the 
transmission loss is probably not more than a 
fraction of a decibel at frequencies lower than 
80 kc. It has been observed that directivity 
patterns for the symmetrical-drive transducers 
housed in cans varied but slightly from the 
patterns obtained from identical crystal motors 
mounted in cylinders of qc rubber. Owing to the 
ease with which tin cans undergo minor de¬ 
formation to accommodate variations in ex¬ 
ternal pressure, they may be used at great 
ocean depths without difficulty. 

In the early transducers of this type as de¬ 
veloped at UCDWR, provision was made for 
two oil plugs on one end of the can; later, a 
single oil plug was used. In a number of in¬ 
stances these plugs were found to leak. It now 
appears that a great improvement in this re¬ 
spect is obtained by providing an approximately 
ViG-m. diameter hole in one end of the can for 
oil filling. Following the evacuation of the can 
it is filled with DB castor oil and then this 
small hole is permanently sealed with a drop of 
solder. This procedure obviates the use of a 
standard oil plug and gives a simpler and more 
dependable seal. Electric leads to the crystal 
motor enter through metal-glass seals of the 
Sperti or Stupakoff type (see Section 8.8.6), 
which have been soldered into one end of the 
can lid. The sealing of the lid onto the can is 
done by means of commercial sealing machines. 
These are obtainable in small hand-operated 
models, which are well suited to experimental 
laboratory use. 

A further discussion of one such transducer, 
UCDWR-type CY4, appears in Chapter 6 and 
a photographic illustration in Figure 22 of 
Chapter 6. A drawing showing construction is 
reproduced in Figure 59. 


Rubber Windows and Cases 
Types of Rubber 

The physical properties of rubber which are 
important to the acoustic performance of trans¬ 


ducers have been discussed in Section 3.7.3. It 
was pointed out that the density and the 
acoustic velocity must individually match that 
of sea water in order that sound waves may 
travel from rubber into sea water without 
suffering reflection or refraction. Samples of 
rubber which meet these two specifications 
have been referred to as qc rubber or sound 
rubber. In some transducers the window shape 
may necessitate the use of qc rubber if the 
window is not to interfere with the acoustical 
performance of the crystal motor, especially its 
directivity pattern. In transducers having flat 
rubber windows of uniform thickness an accu¬ 
rate impedance match to sea water is usually 
not necessary. Instead of oc rubber, it may be 
preferable to use neoprene or some other type 
which has more favorable mechanical prop¬ 
erties. 

Three different types of oc rubber have been 
made available. One, produced by the B. F. 
Goodrich Company, has been very widely used 
in the construction of underwater sound equip¬ 
ment. It carries their designation No. 79-SR-32. 
Its exact composition is unknown to the writer 
since it comes in the category of trade secrets. 
However, it is apparently of natural crude 
stock which has been very heavily loaded with 
castor oil. All samples have a very marked oily 
appearance and also an oily feel. Over long 
periods of exposure part of this oil is lost, at 
least from the surface layers. The rubber is 
quite soft and subject to tearing and eventually 
checks rather badly. In order to bond Goodrich 
QC rubber to metal it is necessary to interpose 
another type of rubber referred to in the trade 
as “tiegum.” This will be discussed further on 
in this section. 

A second type of oc rubber has been made 
available by BTL. Their sound transparent 
rubber. Compound M-163, has the following 
formula. 


Smoked sheet 
Sulfur 
Zinc oxide 
Captax 
Stearic acid 
Heliozone 
Neozone D 
P 33 Black 

Uncured rubber stock 


100.00 parts 
3.00 parts 
5.00 parts 
0.50 parts 
0.50 parts 
2.00 parts 
1.00 parts 
0.50 parts 

having this composition 


^E^TRICTE^ 





HOUSINGS AND ACCESSORIES 


337 


should be cured 30 min at 287 F for sheets 
0.075 in. thick. Test data submitted by BTL 
indicates that rubber of this composition has a 
specific gravity of 0.975, a Shore A hardness of 
35, and a tensile strength of at least 2,800 psi. 
A number of satisfactory transducer windows 
and cases at UCDWR have been made using this 
formula. Some of these were manufactured in 
commercial establishments; others, using un¬ 
cured sheet stock obtained commercially, have 
been made in the UCDWR Transducer Labora¬ 
tory (see Figure 68). As is clear from the 
formula, this compound is primarily crude natu¬ 
ral rubber and contains no castor oil. 

A third type of pc rubber has been com¬ 
pounded at XRL and carries the number F9-5. 
This particular brand of pc rubber is slate gray 
in appearance and is extremely oily. Although 
its exact composition is not known to the writer, 
it appears to be a crude rubber stock containing 
a very high percentage of castor oil. A few 
transducer cases having this composition were 
constructed for UCDWR through the courtesv 
of XRL. 

Another rubber compound that is being used 
currently for underwater sound equipment is 
manufactured by the B. F. Goodrich Company 
and is known as Compound 8388. Acoustical 
data for this type of rubber as well as for many 
other kinds will be found in Table 4 of Section 
3.8. It will be noted that Compound 8388 has 
an acoustic velocity close to that of sea water 
but that its specific gi'avity is 1.15. WTiere a 
strict pc match is not necessary, as in many flat 
windows, this compound may be useful in that 
it possesses superior abrasion resistance. Some 
additional information on types of rubber use¬ 
ful in transducers will be found in Section 3.8. 

Acceptanxe Test 

In order to perform satisfactorily as a win¬ 
dow for acoustic radiation, rubber must be free 
from small air pockets. Even when these air 
cavities possess microscopic dimensions, they 
may still be troublesome from the standpoint 
of acoustic transmission. Although some use 
has been made of X rays in attempting to ana¬ 
lyze rubber windows, it would seem that X-ray 
techniques would not be satisfactory for this 
purpose. 


The best test would appear to be one employ¬ 
ing acoustic radiation. For ease in measurement 
it is desirable to use a very high frequency so 
that the dimensions of the testing equipment 
can be kept small. Equipment designed for such 
tests at XRL employed a frequency of 730 kc 
by using a transducer whose radiating dimen¬ 
sions provided a very sharp acoustic beam. Both 
the sending and receiving equipment were con¬ 
tained in a relatively small tank. 

Rubber Metal Bonds 

In attaching rubber windows to transducer 
cases it is almost always advantageous to have 
the rubber bonded directly to a metal window 
frame, which in turn may be fastened to the 



Figure 66. A cross section of the metal to 
rubber bond in the flat window of the UCDWR 
tjTJe GD case. The metal tongue protruding into 
the rubber results in a longer leakage path. 

main case with a gasket seal. The most impor¬ 
tant exception is the use of cylindrical rubber 
tubes, often referred to in the language of the 
laboratory as “socks,” and discussed in greater 
length in this section and Section 8.9.3. Al¬ 
though the art of bonding rubber to metal is 
quite old it seems to be the consensus that 
rubber-metal bonding must still be regarded 
as an art. Even experts with long experience 
in bonding rubber to metal will occasionally 
produce material that is distinctly inferior or 
even a complete failure. It apparently requires 
meticulous attention to detail. 

There are several known methods for bond¬ 
ing rubber to metals of various types. One of 
the earliest methods required all metals except 
brass to be brass plated previous to the appli¬ 
cation of the rubber. This, too, is a very spe¬ 
cialized technique requiring careful control and 
it would be out of place to discuss it further at 
this point. However, it does seem advisable to 
give a reference to a further source of informa¬ 
tion.-'’ 







338 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


Perhaps one of the simpler methods of secur¬ 
ing successful bonds between rubber and metal 
is to make use of an adhesive known as Ty-ply, 
a trademarked product of the R. T. Vanderbilt 
Company, 230 Park Avenue, New York, Ty-ply 



Figure 67. A cross section of the metal-rubber 
bond in the spherical JK transducer window. An 
extremely long water-to-liquid path is provided 
by the long metal tongue extending into the 
rubber. Note that the bonding of Goodrich oc 
rubber to metal has required an intermediate 
rubber layer; note also the presence of a rectan¬ 
gular shaped rubber gasket for sealing the 
transducer window to its case. 

comes in several formulations intended for 
bonding various rubber compounds to practi¬ 
cally any metal. 

In bonding Goodrich Compound 79-SR-32 to 
metal it is first necessary to follow the coat¬ 
ing of Ty-ply with a layer of intermediate 
rubber or tiegum. The uncured oc rubber stock 
is then placed on the intermediate layer and the 
whole assembly cured together. 


This type of bond is illustrated in Figure 67 
which depicts a cross section of the window for 
a spherical JK transducer. Of especial interest 
in this illustration is the length of path that 
water would have to travel in order to enter 
the transducer along the rubber-metal bond, a 
distance of approximately 5 in. A cardinal point 
in designing rubber metal bonds is to have this 
path as long as possible. It is also important 
to so design these bonds that there will be no 
regions of the rubber subjected to excessive 
tension. Another illustration of the design of a 
rubber metal bond is shown in Figure 66. This 
illustration represents a cross-sectional view of 
a UCDWR-type GD flat window. It will be 
noted that although the total thickness of this 
window is in. the minimum path along the 
bond between the two sides of the window is 
over an inch. 

The Goodrich oc rubber window in the 
UCDWR-type CQ8Z transducer consists of a 
semicylindrical shell 2 in. thick. This window 
has been bonded by butt-jointing the rubber to 
a flat metal surface so that the minimum path 
between the two sides of the window is 2 in. 
(see Figure 33 of Chapter 1). Some difficulty 
has been experienced with the quality of the 
bonds in this particular transducer. It seems 
that zones of maximum stress, which result 
from the marked shrinkage of cured qc rubber 
upon cooling, occur at the external interface 
between the rubber and metal. Some breaking 
away of the rubber from the metal has occurred 
in this boundary region in a number of trans¬ 
ducers, both in and out of service. It now seems 
clear that a superior design for this bond would 
have included a metal tongue which would have 
protruded into the rubber in a fashion anal¬ 
ogous to that which obtains in Figure 67. 

Another example of rubber-metal bonding 
occurs in Figure 68, In this case the rubber has 
been bonded to the metal at both ends of the 
cylindrical cage and also to the steel reinforcing 
rods. The uncured stock. Compound M-163, dis¬ 
cussed in this section, was bonded directly to 
the steel with Ty-ply-Q cement. 

Cylindrical Rubber Cases 

Cylindrical rubber tubes are frequently a 
great convenience in the housing of experi- 























HOUSINGS AND ACCESSORIES 


339 


mental transducers. This is often true even 
though the transducer is intended to radiate 
sound in only one direction. This convenience 
arises from two factors; (1) cylindrical tubing 
is either readily available in stock sizes or it 
can be made to order on short notice from com¬ 
mercial firms who maintain a supply of stock 
mandrels, and (2) a cylindrical rubber tube or 
sock may be advantageously installed by pulling 
it over the metal framework of a transducer 
and then made waterproof by clamping metal 
bands about it at each end. Methods of clamping 
such cylinders are discussed in some detail in 
Section 8.9.3. 

Tubular cylinders of rubber may be formed 
easily in the laboratory with a minimum of 
processing equipment and without the necessity 
of designing expensive molds. To start with, a 
mandrel is prepared whose external diameter 
represents the desired internal diameter of the 
finished rubber cylinder. Upon this cylindrical 
mandrel, which is mounted temporarily between 
centers in a locked position, is wound the un¬ 
vulcanized rubber sheet. Layers of the uncured 
stock, about Yo in. in thickness, are successively 
wound upon the mandrel until the desired di¬ 
ameter is reached. To allow for grinding this 
diameter should be about V 4 , in. larger than the 
outside diameter of the finished product. Ex¬ 
treme precautions must be taken to prevent the 
inclusion of air between the rubber layers. The 
layers of rubber are held in place by wrapping 
the assembly as tightly as possible with thor¬ 
oughly wetted cloth tape. When the rubber is 
later cured in a steam autoclave, great pressure 
is exerted on the rubber owing to the shrinkage 
of the cloth and to the thermal expansion of 
the rubber stock. The correct steam pressure 
and curing time depend on the type of rubber. 
Upon removal from the autoclave, the mandrel 
is again mounted between centers, the cloth 
tape removed, and the rubber cylinder ground 
to a smooth finish of the correct outside diam¬ 
eter. A high-speed lathe tool-post grinder is 
convenient for this operation. 

A slight alteration of this technique which 
results in an improved rubber cylinder for some 
transducer applications, is to bond the rubber 
tube to metal end rings. These end rings are 
machined so that the final seal to the main body 


of the transducer may be made with a gasket 
and gasket groove if desired. The design of the 
metal-rubber bond between the end rings and 
the rubber cylinder should follow the sugges¬ 
tions outlined in this section. 

Molded-Rubber Cases 

Rather extensive use of molded-rubber cases 
has been made, especially for small transducers 
which are cylindrical in cross section. An ex¬ 
ample of a rubber window cap which may be 
included in this category is shown in Figure 21 
of Chapter 1. For the most part, however, these 
cases consist essentially of a somewhat longer 
cylindrical tube with a molded bottom. They 
may be attached to transducers by the banding 
operation described in Section 8.9.3 or they may 
have a metal ring bonded to one end which 
permits a gasket seal to be made to the main 
body of the transducer. 

While some rubber housings may be made by 
cloth-wrapping cylindrical mandrels in the man¬ 
ner described in Section 7.4.4, it is usually 
preferable to mold them under high pressure 
in order to render them free of occluded air. 
Flexibility of design with respect to shape is an 
important consideration, especially for stream¬ 
lining small transducers. 

Steel-Reinforced Rubber 

Transducer cases and windows in which the 
rubber has been reinforced by steel rods or bars 
have been found especially useful in the design 
of inertia-drive units where the crystals may 
be bonded directly to the rubber. This type of 
construction has been discussed in Section 8.7.3. 
The steel-reinforced rubber case for the trans¬ 
ducer shown in Figure 40 of Chapter 1 is illus¬ 
trated in Figure 68. In this transducer the 
location of the crystals is on the interior surface 
of the rubber cylinder midway between ad¬ 
jacent pairs of steel rods. A somewhat similar 
type of construction has been used in the 
UCDWR-type GD 34Z transducer window. Pho¬ 
tographs and drawings of this window appear 
in Chapter 6. Instead of rods it contains rectan¬ 
gular steel bars 1 by % in. in cross section. 

Experience at NRL has shown that unusually 
strong, large sonar domes can be made of rub¬ 
ber by molding into the rubber a lattice work 





340 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 



found that the most satisfactory coating for 
inhibiting corrosion in sea water is a plastic 
material sold under the trademark of Amercoat, 
a product of the American Pipe and Construc¬ 
tion Company, Los Angeles 54, California. 
Amercoat is available in a number of types and 
colors. For best results the treatment consists 
of a priming coat, a body coat, and a seal coat. 
In order to be most effective it is quite important 
to apply Amercoat plastic coating in strict com¬ 
pliance with the detailed instructions furnished 


Figure 69. A UCDWR type CQ transducer with 
its cylindrical rubber window covered with 
marine growth. 

by the manufacturer. Hence, it seems unneces¬ 
sary to enter into a further discussion of 
method at this point. 

Transducer cases constructed of cold rolled 
steel (see Figure 33 of Chapter 1) have been 
painted with Amercoat at UCDWR in order to 


of reinforced steel. By welding a meshwork of 
YiQ-in. steel rods together on 1 1 / 2 -in. centers an 
unusually strong structure can be fabricated 
and yet give little interference to sound radi¬ 
ation at frequencies of 24 kc and less. In order 
to obtain high transmission, the rubber molding 
and the bonding to metal must be done in such 
a manner as to avoid the inclusion of air in the 
rubber. This requires the use of very high pres¬ 
sure during the molding operation. 


Figure 68. The reinforced rubber window used 
with inertia-drive transducers such as shown in 
Figure 40 of Chapter 1. Note the provision for 
capping the ends of this transducer by means of 
the grooves for 0-ring hydraulic gaskets. 

It appears to the writer that the use of rein¬ 
forced steel in rubber windows for underwater 
sound applications has great possibilities and 
that much work should be devoted to its further 
development. 


Corrosion-Resisting Coatings 

The corrosion resistance of metals and alloys 
has been discussed in Section 8.8.1 in connec¬ 
tion with Table 2, which listed these materials 
in an electromotive or galvanic series for sea 
water. Other things being equal, it would seem 
highly desirable to make use of one of the more 
highly resistant metals in the fabrication of 
transducer cases. Since this may not always be 
feasible, for economical or mechanical reasons, 
recourse must be had to methods of improving 
the corrosion resistance of the metals available. 

In recent experience at UCDWR it has been 












HOUSINGS AND ACCESSORIES 


341 


make experimental observations of its effective¬ 
ness. A period of exposure to sea water of ap¬ 
proximately 3 months has elapsed at this writ¬ 
ing and the cases are still in excellent condition. 
There is reason to believe that Amercoat would 
also protect other metals, including aluminum. 
Although plastic coatings seem to offer the 
greatest promise at the present time as a cor¬ 
rosion resistant treatment, additional observa¬ 
tions will have to be made over much longer 
periods of time. 

The fouling of rubber windows by marine 
organisms is a problem of great concern in the 
construction of underwater sound equipment. 
An example of the appearance of a qc rubber 
window that has been badly coated with marine 
life during an exposure to sea water in the San 
Diego area for 3 to 4 months, is furnished by 
the photograph of a UCDWR-type CQ6Z trans¬ 
ducer in Figure 69. Experimental measure¬ 
ments have shown that the transmission of 
sound radiation through such a window has 
been very materially reduced. This decrease in 
transmission has been ascribed not only to the 
presence of organisms themselves but also to 
the gas bubbles entrapped. The directivity pat¬ 
terns of the lobe suppressed receiving array in 
this transducer were very badly distorted owing 
to the marine growth. 

Antifouling paints which are sufficiently flex¬ 
ible to be used on rubber have been under in¬ 
vestigation at NRL. Their formulation NRL- 
P-10 antifouling paint is reported to prohibit 
successfully the attachment of marine growth 
to rubber windows. This paint has been made 
available commercially by the Akron Paint and 
Varnish Company of Akron, Ohio. It is under¬ 
stood that specifications covering this anti¬ 
fouling paint and its method of application 
are contained in BuShips specification No. 
72Re78Z1149A. There has been no experience 
at UCDWR on the effect of this paint on trans¬ 
ducer windows, either acoustically or biologi¬ 
cally. 

^ ® ^ Miscellaneous Seals 

Gaskets 

Most transducers are sealed by means of gas¬ 
kets. Although gaskets may be made of any one 


of several different materials the most common 
composition for transducer applications con¬ 
sists of rubber. The quality of the rubber gasket 
where one is interested in a permanent seal is 
a very important matter. Only natural rubber 
stock should be used where transducers must 
be in service over very long periods of time. 
Materials which will take a set eventually upon 
being subjected to pressure will ultimately re¬ 
sult in a failure of the seal. The best practice 
consists in confining the rubber gasket within 
a groove or other enclosure so that it is possible 
to maintain it permanently under high pressure. 
Such a groove or confined space occurs in nu¬ 
merous illustrations in this chapter, including 
Figures 67, 70 to 73, 81, and 82. 

The most common type and size of rubber 
gasket used at UCDWR was a i/^-in. diameter 
rod. This was bought in rolls and cut to length 
for each individual application, as illustrated in 
Figure 81. The standard groove had a width of 
in. and a depth of %4 in. Neoprene gaskets 
have been used with satisfaction although they 
do take a permanent set. They are never re-used 
in case a transducer is opened for repair. 

0-Ring Hydraulic Gaskets 

0-ring hydraulic gaskets have proved them¬ 
selves very convenient for liquid-tight seals in 
transducer cases. An example of one transducer 
which was designed to be sealed with 0-ring 
gaskets is illustrated in Figure 62. The two 
0-ring grooves of rectangular cross section are 
clearly evident in both the top and the bottom 
bulkhead. This unit was designed to fit into a 
cylindrical case consisting of a rubber sleeve 
bonded to metal end rings. The metal rings had 
a smooth interior finish which permitted them 
to slide over the 0-rings in the assembly of 
Figure 62. The metal parts had a clearance of 
a few thousandths of an inch. The external 
appearance of the case for the transducer in 
Figure 62 was identical with that shown in the 
left half of Figure 68, but the rubber was not 
reinforced with rods. The metal ends of this 
cylindrical housing also contained grooves for 
the installation of 0-rings so that the trans¬ 
ducer case itself could be sealed on either end 
into sections of thin walled tubing. The toler¬ 
ances for the grooves in which 0-rings are to 




342 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


be installed are fairly critical. For suitable de¬ 
sign dimensions reference should be made to 
the standard Army-Xavy specifications for 
0-ring hydraulic gaskets or to other appropri¬ 
ate sources of information. Further details on 
the correct installation of 0-rings will be found 
in Section 8.9.3. 

Glass-Met.\l Terminal Seals 

In many transducers a junction box is pro¬ 
vided in which the cable wires may be attached 
to the transducer terminals. It is customary to 




Figlue 70. Three stages in the installation of a 
glass-metal terminal seal in a Dural bulkhead. 

See text for explanation. 

have the junction box contain air at atmos¬ 
pheric pressure. This arrangement is much 
more convenient than one in which the cable 
enters directly into the oil compartment and 
obviates the use of oil-tight cables. The leads 
from the transducer motor are best brought out 
from the oil compartment by means of glass- 
metal seals. Such seals are available commer¬ 
cially from Sperti Incorporated, Cincinnati, 
Ohio, or from Stupakoff Ceramic and Manu¬ 
facturing Company, Latrobe, Pennsylvania, in 
a variety of types and sizes. Extensive data are 
available from the manufacturers on their elec¬ 


trical and mechanical characteristics. In par¬ 
ticular, tables of safe operating voltages as a 
function of relative humidity are essential for 
design. 

In most cases these glass-metal terminals may 
be soldered directly into the bulkhead which 
separates the liquid compartment of the trans¬ 
ducer from the air-filled junction box. Where 
the bulkhead material does not permit solder¬ 
ing, a method has been devised in which an 
0-ring hydraulic-rubber gasket may be em¬ 
ployed. A hole in the metal bulkhead is made 
with its cross section as shown at the left in 
Figure 70, which also shows the 0-ring in place 
and the glass-metal terminal partially inserted 
in the hole. A rim of metal about the hole has 
a beveled contour so that it can be easily pressed 
against the flange of the glass-metal terminal. 



Figure 71. A view of a hollow glass-metal 
terminal seal installed in a Dural bulkhead with 
an 0-ring gasket. A wire lead is first brought 
out through the hole and then soldered to the 
insulated metal tube. 

It has proved very convenient to use a special 
burnishing tool for the installation of these 
terminals. A satisfactory tool is shown in sec¬ 
tion in the central drawing of Figure 70, just 
prior to contacting the metal for the burnishing 
operation. This tool rotates rapidly in a drill 
press which enables pressure to be applied at 
the same time. The final appearance of the 
glass-metal terminal seal in the bulkhead is 
shown in Figure 71. No difficulty has been ex¬ 
perienced with glass-metal terminals installed 


























HOUSINGS AND ACCESSORIES 


343 


in this manner and many thousands of them 
have been used in expendable transducers made 
of Duralumin. In the glass-metal seal shown in 
Figure 71, a small tube comprises the conduct¬ 
ing element. In some cases these are especially 
convenient in that the leads from the crystal 
motor may be brought out through these tubes 
and then soldered, thereby eliminating the ad¬ 
ditional length of wire which would be required 
if the soldered connection had to be made be¬ 
neath the bulkhead before the latter was placed 
in position. 

Packing Glands for Cables 

Although several types of cable packing 
glands are in current use, all of them function 
on the same basic principle. A common design 
is illustrated in Figure 76. The essential fea¬ 
ture is that the cable shall pass through a gland 
in which a rubber washer can be compressed 
until a tight seal results between the rubber 
and the cable, and between the rubber and the 
internal wall of the gland body. For best per¬ 
formance the dimensions and clearances pro¬ 
vided for a given size cable are fairly critical. 
For this reason detail drawings and a table of 
dimensions for the cable-gland stuffing boxes 
used at UCDWR are reproduced in Figure 72. 
The most common size of cable employed had a 
diameter of 0.38 in. 

Although the metal parts could conceivably 
be made of any one of several metals or their 
alloys, it has been the practice at UCDWR to 
use brass for the entire assembly. In most in¬ 
stances, the composition of the rubber washer 
is very important. In order to maintain a per¬ 
manent seal, this washer should be made of a 
good grade of gum rubber. Otherwise, it is sub¬ 
ject to decompression with a resultant loosening 
of the seal. It has been found convenient to cut 
the rubber washers from Garlock rubber tubing 
having approximately the inside and outside 
diameters listed in the table of Figure 72. The 
correct amount of tightening of the gland nut 
is a critical procedure. If too loose, water will 
leak past it; if fastened too tightly, there is 
danger of breaking the wires within the cable. 
In the limit the entire cable could be pinched 
in two. To start with, threads on packing gland 
nuts should permit the nut to turn very freely. 


in other words, a loose fit is desirable. The 
object is to make it possible to gauge the extent 
to which the rubber is compressed by the torque 
required in turning the nut. Some experience 
in this connection seems to be an essential. 

In some laboratories it is customary to pre¬ 
pare specially the cable at the point where it 



SIZE 

♦ 

A 

B 

C 

D 

E 

F 

G 

H 

j 

K 

L 

M 

N 

CABLE 

GLAND 

STUFFING 

BOX 

4 

1 

5 

8 

z 


'i 

3 

8 

IZ 

64 

3 

4 

7 

8 

5 

8 

1 

J_ 

16 

16 

1 

5 

8 

1 

2 


•i 

3 

8 

21 

64 

3 

4 

7 

8 

1 

8 

1 

4 

X 

4 

(6 

.380 

1 

1 

i 


'5 

1 

H 

i 

7 

I 

1 


16 

1 

'i 

11 

16 

2 

t 


1 

8 

17 

32 

1 

1 

1 

7 

8 

J_ 

14 

5 

6 

'i 

11 

16 

T 

1 

'i 

3 

8 

2L 

32 

> 

, 1 
’e 

f 

1 

J_ 

4 

1- 


Figure 72. Shop drawing of the cable-gland 
stuffing boxes regarded as standard at UCDWR. 


passes through the gland in order to avoid 
damaging the wires by overcompression of the 
rubber washer. This method is presented in 
detail in Section 8.8.8 and illustrated in Fig¬ 
ure 75. 

Oil Plugs 

The present method of installing tapered oil 
plugs at UCDWR has proved quite successful 


































































344 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


and it is very seldom that failures occur in¬ 
volving them. Small plugs with standard iron 
pipe threads are used, preferably of brass and 
usually in a small size as i/4, or % in. Cur¬ 
rent practice calls for carefully cleansing the 
oil plug in benzine and then immediately coat¬ 
ing the clean plug with Glyptal paint before 
any contamination can take place. The concen¬ 
tration of the Glyptal is rather critical. It is 
somewhat too thin as it comes from the can but 
on exposure to the air gradually thickens. When 



Figure 73. Cross section of the oil plug seals 
used by the Bell Telephone Laboratories. Above: 
before tightening. Below: after tightening. 


too thick it does not adhere well, so it is better 
to be too thin than too thick. The recommended 
method of application is complete immersion 
of the threads in the Glyptal. The plug is then 
inserted in the oil plug hole. If the hole has 
been correctly tapped, the plug will begin to 
tighten after a few turns. The old rule, “tighten 
until you can’t turn it any more and then give 
it another full turn,” is not far wrong provided 
care is exercised to avoid stripping the threads. 
When done correctly the Glyptal adheres very 
tightly to both the plug and the case, making 
a very good seal. Units sealed in this manner 
have withstood depths of water up to 800 ft. 

Considerable difficulty had been experienced 
in the past at UCDWR with tapered oil plugs. 
When an oil plug leaked, whether as a result 
of faulty design, defective materials, or im¬ 
proper installation, the crystal motor of the 
transducer often became a complete loss. In 


the early history of UCDWR, the use of tapered 
oil plugs was so unreliable that every unit was 
water tested before it was considered ready for 
service. A pressure chamber designed for use 
up to 300 psi was used as a testing device tc 
detect both defective oil plugs and stuffing 
glands. With electrical attachments on the pres¬ 
sure tank, resistance readings could be made 
continuously on the unit under test. If failure 
was indicated, the pressure was released quickly 
and the unit removed from the water before ir¬ 
reparable damage was done to the crystal motor. 

To avoid the difficulties encountered in mak¬ 
ing watertight seals with tapered oil plugs, 
BTL have adopted a different type of seal. The 
details of their watertight seal are depicted in 
Figure 73. It depends on the compression of 
rubber between two surfaces. Since this type 
of seal has been used satisfactorily for cable 
packing glands and in numerous other applica¬ 
tions, it should be entirely acceptable. The use 
of a cup-shaped spring washer insures a mini¬ 
mum clearance following the tightening of the 
top nut. The rubber washer should be of pure 
gum stock so that it will not take a permanent 
set. After filling the transducer with oil, the 
threaded rod is screwed into the hole; the rub¬ 
ber gasket, the spring washer and the flat 
washer are added in order and the top nut is 
turned down very tightly. 


” ® ‘ Sound Absorbing and Reflecting Pads 
Absorbing Pads 

Materials discussed in Section 8.7.5 provide 
acoustic isolation because they act as good re¬ 
flectors of sound radiation. In many applica¬ 
tions it is highly desirable to absorb the acous¬ 
tic energy. This, however, is quite difficult to 
achieve, especially in a small space. The prob¬ 
lem has been investigated in great detail by 
W. P. Mason and reference may be made to two 
reports from BTL for a discussion of the fac¬ 
tors involved,together with performance data 
on an acoustic measuring tank in which sound 
absorption materials were employed.^i The best 
attenuation reported by Mason resulted from 
the motion of a viscous liquid through small 
interstices in metallic wool pads or in fine-mesh 











HOUSINGS AND ACCESSORIES 


345 


screen. In practice fine-mesh screen is superior 
to metallic wool in that it permits better control 
of the critical dimensions and makes fabrication 
less difficult. Viscous liquids by themselves must 
be used in too great thicknesses to be valuable 
as absorbing layers for underwater sound appli¬ 
cations. 

The screen type of construction is illustrated 
in Figure 74, where some 20 sheets of 100-mesh 
Monel screen, made of 0.004-in. wire, are shown 
separated by strips of coarse expanded metal. 


obtainable from this type of construction, be¬ 
yond which the addition of more screens does 
not result in increased absorption. This practi¬ 
cal limit is in the neighborhood of 30 db for 
radiation reflected from the pad. Some improve¬ 
ment would naturally be expected from replac¬ 
ing the expanded metal by separator strips 
having more nearly the acoustic impedance of 
the liquid; with castor oil, narrow strips of oc 
rubber might be used. To make such a pad 
sufficiently strong mechanically, bonding of the 



Figure 74. Wire mesh attenuator pad developed at the Bell Telephone Laboratories. See text for further 
details. 


On reflection of sound waves from such a pad, 
an attenuation of about 20 db is obtained at 
frequencies over 20 kc. A useful rule-of-thumb 
indicates 0.5 db attenuation per screen for di¬ 
rect transmission or double that for reflection 
losses. A spacing of 5 to 10 screens per inch 
makes a satisfactory construction in the 20- to 
100-kc range. Where special or critical appli¬ 
cations are under consideration, calculations 
should be based on the equations in the articles 
cited^-"’’-^ to determine optimum mesh size and 
spacing. 

There is a practical limit to the attenuation 


metal screen to the narrow rubber strips is 
suggested as highly desirable. Further improve¬ 
ment might also be sought in the direction of 
a better viscous liquid, particularly where tem¬ 
perature fluctuations are large during normal 
usage. Possibly some of the fluids used in hy¬ 
draulic drive mechanisms might be adapted for 
this purpose, such as the Univis oils or the 
Union Carbide and Carbon Company’s series 
HB Ucon 600 oils. The silicones should also be 
considered. In this connection, read also Sec¬ 
tion 8.8.9. 

Since this type of attenuation pad is neces- 














346 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


sarily bulky, its usefulness in small transducers 
is severely limited. The principal application to 
date has been in large acoustic domes in order 
to decrease interference from propeller noises 
and in the measuring tank already mentioned. 
Some of the large transducers might conceiv¬ 
ably profit by these pads. They have been used 
in one transducer designed by BTL, as illus¬ 
trated in Figure 11 of Chapter 1. 

Reflecting Pads 

The fact that it has been difficult to secure 
adequate sound absorbing materials for trans¬ 
ducer applications has led to the wide use of 
reflecting materials. Sound reflecting pads may 
be made of the acoustic-isolation materials dis¬ 
cussed in Section 8.7.5. The most commonly 
used substance is either Corprene or cellular 
rubber. Since they are available in large sheets, 
they may be cut to size readily for covering 
areas of any size and shape which occur inside 
transducers. 

In the transducer shown in Figure 4 of Chap¬ 
ter 6 the interior walls of the steel cylinder 
were lined with Cell-tite rubber. Since this par¬ 
ticular transducer contained both a transmit¬ 
ting and a receiving unit it was also necessary 
to provide for acoustic isolation between them. 
This isolation was provided in part by the in¬ 
sertion of reflecting materials between the two 
crystal assemblies. The presence of the reflect¬ 
ing pads between the two units was not enough 
to eliminate crosstalk entirely since some trans¬ 
mission of sound occurred through the neoprene 
window. When substitution of qc rubber pro¬ 
vided a better impedance match to the sea 
water, there was a noticeable decrease in cross¬ 
talk. In the later production model of this trans¬ 
ducer (see Figure 33 of Chapter 1), the 2-in. 
thick oc cylindrical window had a steel member 
embedded in it to reduce the crosstalk through 
the rubber window. 

The installation of sheets of either corprene 
or cellular rubber is accomplished simply by 
cementing the sheets to supporting structures. 
In the case of Cell-tite rubber it is important 
that the material not be stretched in applying 
it to an extended surface. Oftentimes, these re¬ 
flecting pads may merely be laid in place or 
packed beneath the backing plate. 


Electric Cables 

Specifications and Tests 

Electric cables for transducer applications 
must, generally speaking, satisfy two require¬ 
ments, one electrical and the other mechanical. 
The electrical properties which are primarily 
important have been discussed at length in 
Section 5.2 of Chapter 5. 

Mechanical considerations with regard to 
transducer cables have to do with ultimate ten¬ 
sile strength and freedom from liquid leaks. In 
short lengths of cable the mechanical strength 
of commercial materials is usually more than 
ample. Where extremely long cables are used, 
perhaps 500 or 1,000 ft in length, ordinary rub¬ 
ber covered cable may fail. For such applica¬ 
tions it may be necessary to use cables with steel 
cores. 

The most commonly used cable at UCDWR 
has been a two-conductor shielded cable known 
as Simplex 9061 (also AA60 or SAGO). Its ca¬ 
pacitance per foot and its power factor as a 
function of frequency have been given in Fig¬ 
ure 3 of Chapter 5. 

Oil-Tight Cables 

In transducers where the cable enters directly 
into the oil compartment, it is essential to pro¬ 
vide an oil-tight cable. Although this may be 
accomplished in any one of a variety of ways, 
the most desirable procedure appears to be one 
employed at NRL. This consists in stripping 
off the insulation so that the conducting wires 
are completely laid bare, including the indi¬ 
vidual strands. The strands of wire in each con¬ 
ductor are then fluxed and soldered together to 
give a compact leakproof bundle. The conduct¬ 
ing wires, which protrude about 2 in. from the 
insulated part of the cable, are now placed in a 
special mold, as illustrated in Figure 75. The 
rubber insulation of the cable next to the bare 
wires is now very carefully cleaned and rough¬ 
ened in order to insure that a good bond can 
be made to it with the uncured rubber which is 
about to be placed around the bare wires. Un¬ 
cured rubber stock is now used to fill up the 
mold and the entire assembly is placed in a 
steam autoclave. Curing is done at the pressure 
and for the time recommended for the partic- 


5estricted\ 






HOUSINGS AND ACCESSORIES 


347 


ular grade of rubber in question. Ty-ply-Q (see 
Section 8.8.4) is used on the bare wires and also 
on the rubber cable insulation in order to secure 
adequate bonding. 



Figure 75. Stages in the preparation of a 
cable to make it liquid-tight by molding uncured 
rubber about stranded conductors. Above: Bare 
strands exposed and rubber sheath roughened. 
Below: Strands soldered together and cable 
placed in semicylindrical mold ready for addi¬ 
tion of uncured rubber stock. 



Figure 76. Cutaway view of the cable gland 
seal regularly employed at UCDWR. For stand¬ 
ard dimensions, see Figure 72. 


« Filling Liquids 

Characteristics and Specifications 

The traditional liquid for filling underwater 
sound transducers is Baker’s DB-grade of cas¬ 
tor oil, a highly purified product prepared for 
electric capacitors by The Baker Castor Oil 
Company, Bayonne, New Jersey. The specific 
factors which enter into the choice of this par¬ 
ticular liquid are not a matter of record as far as 
the writer is aware, but its selection could con¬ 
ceivably rest on several properties. Its imped¬ 
ance is a fairly close match to the impedance of 
sea water, castor oil having a density of 0.95 to 


0.96 g per cu cm and an acoustic velocity of 
1,540 m iDer sec. These values may be compared 
to a density of 1.03 g per cu cm and a velocity 
of 1,500 m per sec for sea water. DB castor oil 
is inert toward the many common components 
of a transducer, namely, ADP and RS crystals, 
natural and synthetic rubber, various types of 
adhesives, and many metals. It is relatively easy 
to dehydrate and to free from dissolved gases. 
However, the variation of the viscosity of castor 
oil as a function of temperature is very marked 
and an improvement in this respect could be 
obtained by adopting any one of several other 
liquids. 

A systematic attempt to obtain a better liquid 
for filling transducers has been made by W. P. 



Figure 77. Viscosity-temperature curves for 
various liquids. 


Mason and reference is made to his reporU^ for 
a complete discussion. Mason was interested 
particularly in increasing the power handling 
capacity of crystals and to this end he wished 
to have a liquid with a high cavitation level. 
Among the numerous liquids and vegetable oils 
invesigated, dimethyl phthalate, olive oil, pea¬ 
nut oil, and sperm oil were definitely superior 
with respect to cavitation. With DB castor oil, 
RS crystals were nearly always destroyed by 
burning before any cavitation occurred. For 
increased power handling capacity and higher 
efficiency, the most desirable liquid to use ap- 









































































348 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


pears to be one with a low viscosity. Olive oil, 
for example, which has a viscosity less than 
that of castor oil, can consistently radiate as 
much as 2 w per sq cm without cavitation. This 
may be compared to a reported value of 0.7 w 
per sq cm for castor oil. However, olive oil 
would necessitate that the window be not oc 
rubber, which it attacks, and that a compatible 
cement, such as Acryloid, be used. Dimethyl 
phthalate will likewise stand 2 w per sq cm 
without cavitation, but this liquid softens Vulca- 
lock and Bakelite BC-6052 cement, although it 
does not attack rubber. 

An acetylated castor oil known as Baker’s 
grade P-8 has some desirable characteristics for 
use in transducer construction, but it has been 
found to deteriorate other transducer compo¬ 
nents, especially rubber. In a controlled study 
at UCDWR, it was found that neoprene swelled 
very badly when immersed a few days in the 
P-8 oil. This observation may be contrasted with 
the behavior of neoprene in the presence of DB 
oil, in which case no deterioration has been 
observed even after an exposure of 4 years 
duration. With neoprene cellular rubber stock 
in P-8 oil, it was found that the cellular spaces 
were penetrated and filled with oil after a month 
or two, thereby losing their value as an acoustic 
reflector. This experience conforms to that com¬ 
municated in a letter to UCDWR from the 
Baker Castor Oil Company on July 10, 1945, in 
which they reported the effect of numerous 
liquids on a sample of rubber believed to have 
a natural-rubber base. The per cent increase 
in volume reported for P-8 oil under their test 
conditions was 28 compared to a value of 1 for 
A A grade castor oil (composition reported to 
be practically identical to DB grade). 

Another liquid investigated for transducers 
by NRL is Ucon oil 50-HB-100, developed by 
the Union Carbide and Carbon Company. This 
material is noncorrosive to rubber and bakelite. 
Its impedance approaches that of castor oil and 
it is better from the standpoint of cavitation. 
This oil takes up water quite readily so that 
great care must be taken to see that it is kept 
dry. Dehydration offers some difficulty. When 
purchasing, it is important to insist on material 
which has a very low value of conductivity. Its 
electrical conductivity rises upon oxidation to 


the point of becoming entirely useless, hence it 
must be protected from exposure to air. 

Most promising of all as a transducer liquid 
are the two Dow-Corning fluids, type 200 and 
type 500. The freezing point of all these fluids 
is below —45 C, so that no concern need be had 
in this regard for the normal range of operation 
of sound equipment, even for topside mounting 
on submarines. In fact, some of these liquids 
have freezing points as low as —86 C ( — 123 F). 
The particular characteristic of importance for 
high-powered transducer operation is the com¬ 
paratively slight change in viscosity as a func¬ 
tion of temperature. This characteristic will be 
brought out clearly by an inspection of the 
curves in Figure 77, where graphic data for 
DB castor oil and various other liquids are 
given. These fluids do not deteriorate or soften 
natural rubber, synthetic rubber, or any of 
several types of plastic coating. Whether they 
have an influence on the cements hitherto used 
in transducer construction has not yet been 
tested. They are insoluble in water and the 
low'er aliphatic alcohols but are soluble in most 
organic solvents. Their dielectric constant is 
approximately 2.8 over a frequency range of 
10-^ to 10* c. In fact, the only known deterrent to 
their use at the present time is an economical 
one in that these fluids cost approximately 6 dol¬ 
lars per lb. However, it may be only a question 
of time until these production costs are mate¬ 
rially reduced. These Dow Corning fluids are 
generally known as silicones. They are polymers 
composed of various combinations of organic 
radicals with silicon oxide. 

Dehydration 

It has already been pointed out that owing to 
the solubility of RS and ADP crystals in water, 
dehydration of the liquid used for filling trans¬ 
ducers is necessary. The design of equipment 
for this purpose is an engineering problem 
whose detailed solution may take many forms. 
Without making any claims for the superiority 
of the equipment used for this purpose at 
UCDWR, it will be discussed in order that the 
salient points may be better emphasized. A 
schematic diagram showing the essential parts 
of such a system appears in Figure 78. 

In dehydrating a liquid with a viscosity as 



HOUSINGS AND ACCESSORIES 


349 


high as that of castor oil, difficulty is experi¬ 
enced in securing adequate dehydration in a 
limited period of time. To hasten the removal 
of water, the liquid should be heated to about 
90 F or higher. In addition it is essential to 
decrease the path which water vapor must tra¬ 
verse in order to escape from the body of the 
liquid. This can be done in either one of several 
ways. In the equipment shown in Figure 78 
the castor oil is pumped to the top of a long 


observed emerging from the liquid while still 
subjected to a vacuum, the castor oil is regarded 
as sufficiently dry. While no specific limits on 
the amount of moisture are specified, it can be 
noted that the average moisture content of DB 
castor oil is listed by the manufacturer as 0.01 
per cent and the maximum as not over 0.02 per 
cent. However, there is no possibility of over¬ 
doing the dehydration process since ultimately 
the oil must take up the small amount of mois- 


OIL OEHYORATINS AND OE- 


VACUUM MANIFOLD AERATING TANK 



Figure 78. Schematic diagram of the UCDWR equipment for degassing and dehydrating castor oil; also 
the system used for evacuating and liquid-filling transducers. 


spiral ramp from where it flows down in a thin 
sheet in an evacuated chamber and recirculates 
in this manner until the desired dehydration re¬ 
sults. The dehydration system employed at the 
Brush Development Company achieves a sim¬ 
ilar aim by having the oil fall on rapidly rotat¬ 
ing disks on which it spreads out in a thin 
layer and is then thrown off the whirling disk 
against the wall of the vacuum chamber. At 
BTL, dehydration is achieved by permitting 
small quantities of dry nitrogen at low pressure 
to bubble up through the castor oil contained in 
a series of 3-gallon bottles. 

DB castor oil as purchased is not considered 
sufficiently dry for the direct filling of trans¬ 
ducers. It is usually necessary to circulate it 
for a period of at least 8 hr in an evacuation 
system with a pressure of 1 cm of mercury or 
less. When no further bubbles of vapor are 


ture adsorbed in various parts of the transducer 
or its case and still remain sufficiently dry. 

The glass top which covers the dehydrating 
tank in Figure 78 is regarded as a very desira¬ 
ble feature in that it permits direct observation 
of the condition of the oil. When the vacuum 
pump is initially started, following the intro¬ 
duction of a fresh sample of oil, intense foam¬ 
ing is likely to occur. This foaming can be held 
within proper limits by controlling the pressure 
while observing the behavior of the oil. While 
foaming has always been observed, even with 
new castor oil, it is likely to be much more pro¬ 
nounced in oil which has been reclaimed. 

Where reclaimed castor oil is to be used 
again, as is often done in an experimental 
laboratory, it will be found desirable to incor¬ 
porate a filter system into the oil processing 
equipment. Although no specific brand of equip- 















































350 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


ment can be recommended for this application, 
it is suggested that large oil filters of the type 
employed in commercial motor trucks should be 
satisfactory for this purpose. 

For heating the castor oil, immersion type 
strip heaters have proved convenient. It has 
been the practice to heat castor oil to a temper¬ 
ature of 90 F, although there seems no sufficient 
reason why it could not be dehydrated at an 
appreciably higher temperature. 


8^ FINAL ASSEMBLY AND INSPECTION 
Installation of Arrays 

In the installation of crystal assemblies of 
the backing plate type into the transducer hous¬ 
ing, provision should be made for both mechan¬ 
ical and acoustic isolation. In addition, it may 
be possible to prevent undesired modes of vibra¬ 
tion in the backing plate by a careful consider¬ 
ation of the manner in which it is supported. 

Although well-designed crystal arrays are 
quite rugged, yet some measure of protection 
for the assembly should be provided. It is sug¬ 
gested that a shock mounting be used when 
installing the assembly in its case. Such mount¬ 
ings are usually of a very simple type, often 
consisting of some type of rubber washer. In 
the UCDWR-type GD case illustrated in Fig¬ 
ure 81 the backing plates are usually allowed 
to rest on a number of sheets of Corprene. By 
adding additional Corprene around the sides of 
the motor all metallic connections to the case 
are avoided. When the window is attached, it 
presses against a wide rail on the backing plate 
(see Figure 34 in Chapter 1) and holds the 
crystal motor in place. The radiating face of 
the crystal array is usually placed a short dis¬ 
tance back of the window, perhaps Vr of an 
inch. In packing Corprene in the type-GD case, 
channels must be provided to facilitate evacua¬ 
tion of the case and the subsequent liquid-filling 
operation. 

The method of supporting backing-plate ar¬ 
rays with mounting brackets will be clear from 
Figure 49. The attachment of these mounting 
brackets to a transducer case can be visualized 


by referring again to Figure 4 of Chapter 6 or 
to Figure 33 of Chapter 1. 

According to the experience of BTL in 
mounting the crystal array shown in Figure 50, 
improved performance was obtained by attach¬ 
ing the supporting brackets some distance from 
the corners of the steel backing plate. This 
slight alteration in the point of support seemed 
to be effective in suppressing undesirable modes 
of vibration in the steel backing plate. 

The problem of mounting inertia-drive crys¬ 
tal arrays solves itself for those units where 
the crystals are bonded directly to a rubber 
window. Examples of this type of mounting 
are to be seen in Figure 56 of this chapter and 
in Figure 40 of Chapter 1. The method of in¬ 
stalling the inertia-drive window unit of Fig¬ 
ure 56 in its case is illustrated in Figure 82 
and discussed in Section 8.9.3. With inertia- 
drive units that are not bonded to windows 
some other provision must be made for mount¬ 
ing them in a case. An illustration of one such 
unit is shown in Figure 62, in which instance 
the assembly is inserted in a cylindrical rubber 
housing and sealed by means of an 0-ring gas¬ 
ket. This process is discussed in somewhat 
greater detail in Section 8.8.6. 

With stack-type crystal assemblies as devel¬ 
oped at UCDWR up to the present time, their 
installation into a proper housing has been an 
exceedingly simple procedure. The stack unit 
illustrated in Figure 59 has circular disks of 
Corprene or rubber attached to each end of the 
crystal array, which center the assembly within 
a cylindrical tin can. These Corprene or rubber 
disks have the same diameter as the interior of 
the can and their thickness is selected so that 
they press lightly against either end of the can. 
It is usually also desirable to have the corners 
of the assembly, which in Figure 59 consist of 
Corprene and Cell-tite rubber, press lightly 
against the interior wall of the can. 

The stack assembly in Figure 60 was mounted 
in its case by cementing thick rubber disks on 
either end of the unit and supporting the array 
between two rigid bulkheads, which were them¬ 
selves attached to either end of a cylindrical 
rubber sock. However, to obtain sufficient 
strength in the housing and still permit radia¬ 
tion over a 360-degree angle, it was necessary 



FINAL ASSEMBLY AND INSPECTION 


351 


to couple the bulkheads with a cage made of 
expanded metal. In the frequency range of 60 
to 90 kc, the expanded metal did not seriously 
interfere with either the directivity pattern or 
the output level of the transducer. 


^ ^ Matching Networks and Cables 

Information pertinent to the design and con¬ 
struction of matching coils and/or transformers 
for transducers has been discussed at length in 
Chapter 5. In the design of transducer cases 
it is usually desirable to provide a satisfactory 
cavity in which these matching networks may 
be placed. From the standpoint of installation, 
few difficulties are likely to be encountered. Of 
principal concern is the necessity for securing 
adequate electrical insulation for the fairly high 
potentials to which these networks are sub¬ 
jected during operation. Where the tuning coil 
or the transformer is immersed in the trans¬ 
ducer liquid, insulation is a very simple matter; 
for location in an air cavity, these components 
must be provided with the required insulation. 

Perhaps the most common difficulty at 
UCDWR in connection with the installation of 
matching networks had to do with mistakes in 
connecting the leads of the network to the 
proper terminals of the transducer. Errors of 
this sort usually resulted from an improper or 
insufficient labeling of the leads. Further elab¬ 
oration of this point need not be made since 
the remedy is well-known. 

Since the inductance of either tuning coils or 
other matching network depends on the spatial 
relationships between the various windings and 
also on the proximity of the windings to neigh¬ 
boring metallic boundaries, special precautions 
may need to be taken at times, especially with 
uncased network components, in order to retain 
or attain the values of inductance called for in 
the specifications. In transducers designed for 
quantity production, this difficulty would prob¬ 
ably be met by having all of the electrical net¬ 
works permanently cased and provided with 
outside terminals so that the final check on their 
inductance could be made before their installa¬ 
tion in the transducer housing. 

In all except small transducers it has been 


customary to provide a separate compartment 
where the electric terminals of the transducer 
can be attached to the cable. The wire leads 
from the crystal array or from the matching 
network are brought into this compartment 
through liquid-tight seals. This terminal com¬ 
partment is usually air-filled in order to obviate 
the leakage of oil, which might otherwise occur 
through the main transducer cables. Between 
the main oil cavity of the transducer and the 
air-filled compartment, it will be found con¬ 
venient to provide glass-metal terminal seals as 



Figure 79. Tightening the packing gland nut 
on a UCDWR type GD transducer. See text for 
discussion. 


described in Section 8.8.6. Another method of 
providing such a liquid-tight seal would be to 
use a modification of the cable-gland stuffing 
box shown in the shop drawing of Figure 72. 
Where cable is used for this type of connection, 
it is important to remember that the cable itself 
must be made oil-tight, perhaps by the method 
discussed in Section 8.8.8 and illustrated in Fig¬ 
ure 75. It should be remembered also that for 
transducers which operate at great depths the 
differential pressure between the liquid com¬ 
partment and the air-filled terminal-block 
compartment may amount to several hundred 
pounds per square inch. 

In the installation of cable it is necessary to 
avoid manipulating it in such a way as to break 
any of the conductors. In particular one should 
avoid flexing the cable at sharp angles. In tight- 








352 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


ening the nut of the cable-gland stuffing box it 
is possible to damage the cable by exerting too 
great a static pressure on the rubber washer. 
In extreme cases this can even result in break¬ 
ing the conductors and thus causing an open 
circuit. The packing gland nuts should always 
be freely turning so that it is possible to judge 
the proper force necessary to tighten them to 


® Sealing Transducer Cases 

Banding Rubber Cylinders 

The suitability of cylindrical tubing as a 
housing for transducers has been discussed in 
Section 8.8.4. Much of the advantage derived 
from such cylindrical socks results from the 
ease with which waterproof seals may be made 



Figure 80. Stages in wire-banding a cylindrical rubber sleeve on a transducer. After the wire loop (A) 
is placed around the rubber cylinder, its ends are firmly gripped by the clamping tool (B) while the tongue 
of the tool engages the closed end of the wire loop (C). When the wire has been tightened sufficiently by 
turning the hand wheel (C), the wires are bent sharply around the loop (D), the tool removed and the 
wire ends cut down to proper length (E). The wire loop is secured by further bending of the wire ends 
with pliers (F). 


the desired point. In the illustration of Figure 
79, where a 0.38-in. cable stuffing box (see Fig¬ 
ure 72) is being tightened with a wrench, a 
force of 30 lb at a distance of 7 in. from the 
cable is an approximately correct value. A little 
experience will enable one to judge the proper 
torque for satisfactorily tightening these 
glands. 


to the underlying metal case of the transducer 
by a simple banding process. The metal ends of 
the transducer case should preferably possess a 
number of grooves or serrations. When metal 
bands are clamped tightly about the rubber 
cylinder, very high stresses will occur in the 
rubber in the region of these serrations, thus 
resulting in a dependable seal. 










FINAL ASSEMBLY AND INSPECTION 


353 


Cylindrical tubing in smaller sizes may be 
clamped by means of wires, preferably stainless 
steel. The several steps constituting an accept¬ 
able technique for wire banding are illustrated 
in Figure 80. A commercial tool designed for 
this process has been made available by the 
Chicago Pneumatic Tool Company. The proper 
length of wire required for a particular band is 
formed into a loop by bending it at its center 
(Figure 80A). The wire loop is curved about the 
rubber cylinder so that both ends of the wires 
extend through the loop and into holes in the 
special tool where they are clamped in place 
(Figure 80B). The wire band may be tightened 
now by turning the knurled head at the oppo¬ 
site end of the clamping tool until there is a 
marked depression formed in the rubber (Fig¬ 
ure 80C). Judgment with regard to the proper 
amount of pressure on the rubber will be gained 
by experience. When the wire is considered suf¬ 
ficiently tight the tool is forced sharply back¬ 
ward as shown in Figure 80D and a finger 
should be held over the two wires to prevent 
them from unbending when the tool is removed 
with the other hand. While continuing to hold 
down the loose ends of the wires, one of them is 
clipped shorter with side cutting pliers (Figure 
80E) and bent down securely as illustrated in 
Figure 80F. Then the excess length of the sec¬ 
ond wire is cut off and anchored securely in the 
same manner. 

For cylindrical transducers whose diameters 
are greater than 4 in., the Punch-Lok type of 
band, manufactured by the Punch-Lok Com¬ 
pany, Chicago 7, Illinois, is preferred. The 
Punch-Lok bands, which are % in. wide and 
approximately either 0.020 or 0.030 in. thick, 
are available in various lengths and in several 
metals, including stainless steel. It is recom¬ 
mended that stainless steel be employed in in¬ 
struments which are to remain in service under 
water an appreciable length of time, as ordinary 
iron bands would quickly deteriorate. A trans¬ 
ducer utilizing this type of banding is shown in 
Figure 69. A special banding tool is essential 
for proper installation. The procedure involved 
is discussed in the manufacturer’s direction 
sheets in sufficient detail. The only special com¬ 
ment required here is that serrations in the 
metal beneath the clamped region are recom¬ 


mended, as outlined in the previous paragraph 
on wire banding. Again, experience will help in 
estimating the amount of compression required 
in the rubber for a waterproof seal. 

A convenient method of tightening a stainless 
steel band on a transducer is illustrated in Fig¬ 
ures 21 and 23 of Chapter 1. The stainless-steel 
bosses, which have been welded to the bands at 
either end, have been provided with a clamping- 
screw mechanism. 

Gasket Joints 

In sealing transducer cases it has been cus¬ 
tomary to use rubber gaskets to attach various 
lids and metal plates to the housing. Such metal 
surfaces occur where rubber windows have 



Figure 81. Inserting a rubber gasket in the 
rectangular groove of a UCDWR type GD trans¬ 
ducer case so that the squarely cut ends of the 
round rubber rod meet under slight compression. 

been bonded to metal frames, and where cover 
plates containing cable glands (see Figure 33 of 
Chapter 1) are used to seal the electric terminal 
compartment. In the installation of rubber 
gaskets in connection with any of various com¬ 
ponents there are two or three precautions to 
be observed. It has already been indicated in 
Section 8.8.6 that the dimensions of the gasket 
groove are rather critical since the object is to 















354 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


force the gasket into a confined space. Similarly 
sufficient gasket material must be present with 
respect to length to fill the groove adequately. 
Where the Vs-bi. rubber gasket rod is cut to 
length it is recommended that it be cut slightly 
oversize so that the two cut ends will butt to¬ 
gether firmly as depicted in Figure 81. These 
ends may be cut either perpendicular to the 
length or, provided the two ends match, at a 
slight angle. 

In tightening a metal plate down on a gasket 
care must be exercised to see that approxi¬ 
mately equal pressure is exerted along the en- 



1 

Figure 82. Closure of a UCDWR type EP 
transducer (also see Figure 56) with an up¬ 
setting die (above) which is forced down over 
the transducer case by means of an hydraulic 
press, thus curling the thinned steel wall of the 
cylindrical case around an 0-ring gasket. A com¬ 
pleted crimp seal appears on the bottom end of 
the case. 

tire length of the gasket. This means that the 
individual bolts or screws should be tightened 
gradually and across diameters in order to 
avoid nonuniform compression of the gasket. 
An attempt should be made to inspect the 
gasket, if at all possible, during this tightening 
process to see that it has retained its correct 
position. 


If a gasket groove possesses a circular con¬ 
tour whose length permits taking advantage of 
the availability of a wide range of closely 
graded sizes of 0-ring gaskets, it may be prefer¬ 
able to do so instead of cutting a rubber rod to 
the proper length. 

For the large spherical JK-type of trans¬ 
ducer, photographs of which occur in Chapter 1, 
the spherical dome is sealed to the main body of 
the housing by a rubber gasket having a rec¬ 
tangular or square cross section. A cross-sec¬ 
tional view of this gasket appears in Figure 67 
where the hemispherical window is shown 
bonded to the housing. It will be noted that the 
rectangular gasket is closely confined in a cav¬ 
ity near the bolt circle of the window frame and 
that it is compressed between the window frame 
and the flange of the housing. When the window 
is once placed in position it is not possible to 
inspect this gasket visually and a check on the 
tightness of the seal should be made by subject¬ 
ing the transducer to internal air pressure or by 
actually testing it under water. The former pro¬ 
cedure has been recommended, using an air 
pressure of 35 psi or less. 

0-Ring Installation 

In closing transducers which contain 0-ring 
hydraulic gaskets great care must be exercised 
to avoid damaging the rubber 0-ring. It fre¬ 
quently is helpful to apply a small amount of oil 
to the gasket in order that the metal parts may 
slide over it more freely. In assembling trans¬ 
ducers castor oil may be used for this purpose. 
Prior to their installation 0-rings should be ex¬ 
amined very carefully in order to eliminate 
those which contain any imperfections. Only 
perfect gaskets should be used. 

Provision for proper clearance between metal 
transducer parts is a matter for the designer to 
handle. However, it should be pointed out that 
for a transducer seal which constitutes an es¬ 
sentially permanent installation, much less 
clearance should be provided for the 0-ring 
than is specified in the standard Army-Navy 
specifications. This is owing to the fact that the 
specifications were set up for the case of rotat¬ 
ing seals. Additional discussion of 0-ring hy¬ 
draulic gaskets is contained in Section 8.7.4. 
Apart from their use in sealing transducer cases 
of the type shown in Figure 62, they may also be 




































































FINAL ASSEMBLY AND INSPECTION 


355 


used for such applications as those pictured in 
Figures 70 and 82. 

Crimp-Sealing Methods 

In the construction of many thousands of ex¬ 
pendable transducers, it has been found eco¬ 
nomical as well as entirely satisfactory, to ob¬ 
tain a liquid-tight closure by crimping the ends 
of the tubular metal housing. As an illustration, 
the transducer motor shown in Figure 56 can be 
very readily sealed into a cylindrical steel case 
in the manner illustrated in Figure 82. This sec¬ 
tional illustration depicts the approximate wall 
thickness required for a steel tube 3 in. in di¬ 
ameter. Above the transducer a special crimp¬ 
ing die is in position, ready to be pressed down 
against the open end of the transducer. A rub¬ 
ber 0-ring hydraulic gasket is in its place on 
top of the steel rim of the rubber window dia¬ 
phragm. As the die is forced down over the 
transducer case, preferably by means of a hy¬ 
draulic press, the thin steel wall curls around 
the 0-ring. The appearance of a finished seal is 
illustrated in cross section at the opposite end 
of the same transducer case. 

Another type of crimp seal has been discussed 
in Section 8.8.3, where the use of tin cans as 
transducer cases was presented. 

Liquid-Filling Technique 
Evacuation 

There may be two reasons for evacuating 
transducers before filling them with liquid. One 
reason is purely mechanical; with the filling 
equipment employed at UCDWR, it is much 
simpler to insure that the transducer is com¬ 
pletely full of gas-free liquid if all air has been 
previously removed. Moreover, transducers may 
be filled with liquid much more rapidly when 
evacuated. The second reason for evacuating 
transducers is owing to the almost inevitable 
presence of moisture on the surfaces of the crys¬ 
tals. This is especially serious with RS crystals 
and has been discussed in the early sections of 
this chapter, particularly in Section 8.2.4. 

In order to make certain that moisture has 
been removed from the crystals it has been cus¬ 
tomary at UCDWR to connect the terminals of 
the array to an external ohmmeter while the 
transducer is in the vacuum chamber. In this 


manner the d-c resistance of the unit can be 
checked during the pumping process. 

The evacuation of air and moisture from a 
transducer can take place by either one of two 
methods. One method is illustrated in Figure 
78, in which the entire transducer is placed 
inside of a large vacuum tank. The principal 
advantage in this method is that it does not sub¬ 
ject the walls of a transducer to the dilferential 
pressure of 1 atmosphere. This is usually an im¬ 
portant consideration. In transducers which are 
capable of withstanding a differential pressure 
of 15 psi, it is somewhat simpler to provide 
them with two openings so that a vacuum line 
could be connected to one and, at the proper 
time, the liquid could fiow into the transducer 
through the other. 

Filling with Liquid 

Where a transducer has been placed inside a 
vacuum tank for the evacuation process, it may 
be filled with liquid in the manner shown in 
Figure 78. In this equipment the dehydration of 
the castor oil takes place in an adjacent tank, 
which is coupled through the recirculating 
pump to the vacuum tank for this filling opera¬ 
tion. By opening one valve and closing another 
(not shown), the purified castor oil is pumped 
through the oil line into the transducer. It is 
customary to provide a sump as shown so that 
oil may be allowed to run into the transducer 
until it appears in the sump. The object is to 
prevent air from entering the transducer when 
atmospheric pressure is established again inside 
the tank. After filling, the oil hole in the trans¬ 
ducer is sealed according to the directions given 
in Section 8.8.6. 

With transducer housings capable of support¬ 
ing a differential pressure of 1 atmosphere, the 
evacuating and filling operation can be accom¬ 
plished more simply by methods involving a sys¬ 
tem of valves. Further elaboration on such sys¬ 
tems seems unnecessary in a volume of this 
kind. 

8.9.5 Final Inspection and Testing 
Leaks 

One of the troublesome factors in the con¬ 
struction of transducers is the frequent pres¬ 
ence of liquid leaks in the final assembly. The 




356 


CONSTRUCTION TECHNIQUES AND EQUIPMENT 


detection of leaks which may occur in the metal 
casting itself has already been discussed in Sec¬ 
tion 8.8.2. In the case of welded transducers, 
there is a still greater possibility that leaks may 
be present. Another possible source exists in 
bonds between metal and rubber. Leaks of these 
types may usually be investigated before the 
crystal array is mounted in the housing. 

The gasket with which the major opening of 
the transducer is sealed may also be a trouble¬ 
some cause of leaks. The stuffing-box seal for the 
cable gland may likewise be a source of annoy¬ 
ance. The proper procedure for the installation 
of cables has been discussed in Sections 8.8.6 
and 8.9.2. When properly installed by an experi¬ 
enced worker, these cable glands should not fail. 

The closure of the hole for liquid filling has 
been discussed in Section 8.8.6. As this usually 
constitutes the last opening in a transducer 
which is closed, there is normally no method of 
testing it for leaks. This is a fundamental diffi¬ 
culty in transducer design for which a remedy 
should be found. It seems that it should be pos¬ 
sible to develop some type of small testing 
gadget by means of which a final inspection test 
of all sources of leaks could be made. 

Previous to filling a transducer with oil, the 
entire housing and assembly may conveniently 
be tested for leaks by the use of dry compressed 
air, preferably introduced through the oil plug 
hole. After transducers are oil-filled, a test could 
be made by immersion in water over a suffi¬ 
ciently long period to see whether the housing 
actually leaked. This could be determined by a 
continuous d-c resistance measurement. In 
order to accelerate such a leak test, the trans¬ 
ducer may be subjected to a hydrostatic pres¬ 
sure of several hundred pounds per square inch. 
Should leaks be present in the housing accord¬ 
ing to readings taken on the meter, the trans¬ 
ducer may be immediately removed from the 
water and disassembled for repair before the 
crystal arrays are damaged. 

D-C Resistance 

The quality of a transducer most frequently 
subjected to test both during and after con¬ 
struction is its d-c resistance. Not that the d-c 
resistance has an important bearing as far as 
the final operation of the transducer is con¬ 


cerned, but largely owing to the fact that d-c re¬ 
sistance measurements are so easily made and 
yet furnish valuable indications of the quality 
of the construction. Since one is usually not in¬ 
terested in the absolute value of the d-c resist¬ 
ance, nor in the accuracy of its determination, 
the readings may be made on any vacuum-tube- 
type of ohmmeter. The voltage applied by such 
a meter should not be in excess of 500 v for gen¬ 
eral purpose use. 

The d-c resistance to be expected from crystal 
arrays of either RS or ADP was indicated in 
Section 8.7.6. In a completely assembled trans¬ 
ducer which may contain tuning coils or another 
type of matching network in addition to the 
crystal assembly, the d-c resistance may be un¬ 
duly influenced by the network. Where a match¬ 
ing transformer is used, the d-c resistance 
would be expected to be comparatively low so 
that little or no indication of the condition of 
the crystal assembly could be obtained in this 
case by a d-c resistance measurement on the ex¬ 
ternal terminals of the transducer. 

In case a low resistance reading is obtained 
when a measurement is taken directly across 
the terminals of the crystal assembly itself, its 
cause may usually be assigned to contaminated 
crystal surfaces, especially to the presence of 
excess moisture, or to an individual crystal 
which has failed. In the latter event it may be 
possible to chip out the offending crystal with¬ 
out interfering with the behavior of the trans¬ 
ducer as a whole. 

Where d-c resistance measurements on the 
external terminals reveal a short circuit or a 
very low resistance, the cause may lie in any one 
of several directions. A systematic investigation 
of the possible sources which could contribute to 
a low resistance reading should be made. The 
cable should be removed and a check made for 
moisture in the stuffing box and in the terminal 
compartment. If these are satisfactory the ter¬ 
minal block should be tested, one terminal at a 
time, for proper insulation. If any indication of 
water is present, each terminal must be care¬ 
fully cleaned. For transducers that have been 
immersed in sea water, warm water should be 
used for cleaning away the electrolytic deposit; 
then the terminals are dried thoroughly and re¬ 
tested. 


[^STRrCTE4 





FINAL ASSEMBLY AND INSPECTION 


357 


In addition to measuring the d-c resistance 
between the two terminals of a transducer, it is 
also necessary to measure it between each one of 
the terminals and the ground connection; also 
between each individual terminal and the shield 
on the cable. The d-c resistance between the ter¬ 
minals and either ground or shield should be 
very high, the meter indicating anywhere from 
a few megohms upwards. Where low values of 
resistance are found, a thorough check of the 
insulation of the terminals should be made. The 
presence of soldering flux or too high a tempera¬ 
ture during soldering is often to blame. The in¬ 
sulation may need to be carefully washed with 
warm water to remove the flux and then dried; 
or perhaps cleaned with some satisfactory or¬ 
ganic solvent. 

Calibration 

The numerous types of calibration data that 
may be obtained for transducers have been 
listed and discussed elsewhere in this volume. A 
preliminary presentation of the calibration 
measurements that may prove desirable was 
contained in Section 1.3. A more complete treat¬ 
ment of the complex impedance was given in 
Section 4.5 and the various steady-state re¬ 
sponses have been considered in detail in sepa¬ 
rate subsections of Section 4.6. Directivity pat¬ 


terns have been treated in several subsections 
under Section 4.3. Reference should be made to 
these sections for a thorough discussion of cali¬ 
bration subjects. With regard to methods of 
measurement and calibration equipment, it is 
the understanding of the writer that an entire 
volume in the series of Summary Technical Re¬ 
ports is being devoted to them. 

In addition to indicating performance char¬ 
acteristics, calibration data are useful for in¬ 
dicating shortcomings both in design and in 
construction. Directivity patterns in some types 
of transducers are especially sensitive to con¬ 
structional variations and hence serve as a 
check on correct assembly. This is true of stack- 
type transducers. Another example is the effect 
produced by the accidental inclusion of an X-cut 
crystal in a Y-cut RS array. Owing to the low 
impedance of the X-cut crystal, which receives 
most of the power, the pattern may resemble 
that of a point source instead of the Y-cut ar¬ 
ray. 

Impedance measurements are most useful 
from the standpoint of construction in connec¬ 
tion with determining the design values for 
matching networks and in checking their subse¬ 
quent performance. Resonant frequencies of the 
transducer under water are also quickly ob¬ 
tained from impedance data. 




Chapter 9 

RESEARCH TECHNIQUES AND APPARATUS 

By T. Finley Burke, Francis X. Byrnes, and Bourne G. Eaton 


’ * ELECTRICAL MEASUREMENTS 

I N BUILDING AND TESTING crystal transducers 
it is necessary to measure some of their elec¬ 
trical properties. Some of these properties are 
measured by tests that are essentially direct 
current in character, such as the simple push 
test for polarity and activity, and also the d-c 
resistance test. The methods and instruments 
used and the usual range of results obtained in 
making these tests are given in Chapter 8. The 
other electrical properties of great interest are 
the impedance of the transducer and the im¬ 
pedances of some of the electrical components 
used in the transducer. The methods and instru¬ 
ments used in measuring these impedances will 
be described in the following paragraphs. 


Absolute Admittance 

The simplest measure of the impedance of a 
network is a measurement of the absolute mag¬ 
nitude of its impedance. In actual practice, be¬ 
cause of practical considerations in making the 
measurements, the quantity that comes directly 
out of the data is not impedance but the inverse 
quantity, absolute magnitude of the admittance 
of the circuit. The following discussion will 
therefore consider this measurement as an ad¬ 
mittance measurement rather than an imped¬ 
ance measurement. 

A measurement of the magnitude of the ad¬ 
mittance of a completed crystal transducer, 
when it is loaded by the water, has a very lim¬ 
ited use because of the very small changes in 
the magnitude of the admittance that are pro¬ 
duced by significantly large changes in the mag¬ 
nitude of the various mechanical and acoustical 
admittances that are coupled into the crystal 
circuit. This is true because the magnitude of 
the admittances represented by the water im¬ 
pedance, and various other stray admittances, 
remain so low in comparison with the admit¬ 


tance of the purely electrical capacitance of the 
crystal that they exert very little effect. 

If a single crystal is measured in air, how¬ 
ever, the admittance may be quite useful in 
checking many of the crystal’s properties and in 
detecting defects in particular crystals. For ex¬ 
ample, the measured value of the admittance, 
as compared with the calculated value, meas¬ 
ured at some frequency well below the first reso¬ 
nance, can be used to check whether or not the 
crystal has been cut out of the mother crystal at 
the correct angle. This is possible because of the 
fact that small errors in cutting angle will cause 
rather large changes in the capacitance and 
therefore the admittance of the crystal. This is 
especially true of Y-cut Rochelle salt [RS]. 
Other useful quantities that may be determined 
by an absolute admittance measurement are: 
(1) the maximum value of the admittance at 
resonance, (2) the minimum value of the ad¬ 
mittance at antiresonance, and (3) the fre¬ 
quencies at which the resonance and antireso¬ 
nance occur. Using these experimentally deter¬ 
mined quantities in the following equations, the 
components in the first approximation equiva¬ 
lent circuit for a crystal may be determined. 
(See Figure A.) 


Cy Ly 


3- 

- j\ -ObbU- 

Co ^ 

3- 

- Ry ■: 


Figure A. 


2Co(/. - fr) 

fr 

1 

- fr)Co’ 

1 




358 












ELECTRICAL MEASUREMENTS 


359 


also R,, ~ 

^ max 

^ ■ Co - fr 

C,,-2(/„ 

where, Ctotai = Co + Cm (which is the direct 
capacitance measured at a fre¬ 
quency well below the first reso¬ 
nance) , 

fr = frequency of resonance 
(Y = maximum), 
fa = frequency of antiresonance 
(Y = minimum), 

^max = maximmn value of the admit¬ 
tance (/ = fr), 

Ymin = minimum value of the admit¬ 
tance (/ = fa). 


All the approximations given above hold quite 
well for small values of In particular, they 
hold quite well when the crystal is free, in air, 
and is mounted at the center with reasonable 
care to prevent damping. 

Although a knowledge of the values of the 
elements in the equivalent circuit of a single 
crystal in air has a rather remote relationship 
with crystals built into a transducer operating 
in the water, it does furnish a method of de¬ 
termining some of the fundamental properties 
of crystals. By observing the changes in the ob¬ 
served values of the elements in the mechanical 
arm of the crystal circuit produced by attach¬ 
ments, the corresponding admittance measure¬ 
ments may be used to determine the properties 
of various attachments to the crystals, such as 
cement joints and the materials that are at¬ 
tached to the crystal by means of the cement 
joints. 

The circuit used in a determination of the 
absolute admittance of a crystal is shown in 
Figure 1. The requirements of the various com¬ 
ponents in this circuit are as follows: 

The oscillator should preferably have a low 
output impedance so that its output voltage will 
remain reasonably constant as its frequency ap¬ 
proaches that at which the crystal is resonant. 
It must also have very low-harmonic content in 
its output, even when working into the very low 
impedance represented by the crystal at reso¬ 
nance. The maximum value that the first har¬ 
monic may have without causing appreciable 


error will depend somewhat on the circum¬ 
stances but may be taken as approximately 
per cent. This expression will have a 
value as low as 0.1 per cent for quite commonly 
encountered values of Co and R^,. The higher 
harmonics should be even smaller in amplitude, 
falling off at a rate no lower than that at which 
the frequency increases. 

No special characteristics are required of the 
voltmeter used to measure the output of the 


OSCILLATOR 




VOLTMETER 
NO. I 


VOLTMETER 

N0.2 



Figure 1. Circuit used for measuring the abso¬ 
lute admittance of crystal transducers or ele¬ 
ments thereof. 


oscillator. The meter used to indicate the volt¬ 
age across the current resistor R, must be ca¬ 
pable of accurate measurements over a large 
voltage range. This is because the current varies 
widely with frequency in passing from reso¬ 
nance to antiresonance. The voltage range re¬ 
quirements may be lessened appreciably by 
changing the value of the resistor R, in accord¬ 
ance with the change in the measured admit¬ 
tance. For example, when the admittance is 
measured at resonance the current is quite high 
and a small value of R may be used. When the 
admittance at antiresonance is being measured 
the current will be quite small and a much 
larger value of R will be helpful. Since the im¬ 
pedance of the crystal is very high at antireso¬ 
nance the higher value of R may be used with¬ 
out introducing error caused by the voltage drop 
across it. It should be noted that the value R 
should never exceed one-tenth of the impedance 
1/Y that is being measured. This is necessary 
because a valid correction for a large value of R 
cannot in general be made as one has no knowl¬ 
edge of the phase angle of Y. 

The only requirements placed upon the other 
elements of the circuit, the wiring and the re¬ 
sistor, are that they contribute no reactance 
comparable to that of the crystal circuit even 
when the crystal is near resonance or antireso¬ 
nance. 




















360 


RESEARCH TECHNIQUES AND APPARATUS 


^ ^ ^ Complex Impedance of Two- and 
Three-Terminal Networks 

A measurement of the absolute value of the 
electrical admittance is of great value in deter¬ 
mining some properties of crystal transducers 
and of the components within the transducer, 
but it cannot yield all of the information which 
is required in the case of transducers which are 
radiating into water. Thus loaded, the resistive 
component is so large that there is very little 
change in the absolute value of admittance. 
However, bridge measurements of the real and 
imaginary terms of the admittance or imped¬ 
ance do give the necessary information under 
these conditions. The measurements are more 
tedious and it is usually advisable to first make 
a general survey by the absolute method and 
then conduct a more detailed study of the in¬ 
teresting regions with the aid of an impedance 
bridge. 

With an ordinary impedance bridge two- 
terminal impedances may be measured, i.e., the 
unknown impedance has two terminals which 
are connected to the bridge, and the impedance 
between those terminals is measured. In the 
case of a crystal transducer this measurement is 
not always sufficient. Such two-terminal imped¬ 
ances are sometimes of value in predicting a 
transducer’s performance when connected into 
a circuit, but such measurements can tell us 
very little about the arrangements of stray ca¬ 
pacitances within the transducer. In some cases, 
these inactive capacitors together with their 
dissipation might actually be consuming a large 
part of the power being delivered to the trans¬ 
ducer. In attempting the development or im¬ 
provement of transducers it is very important 
to recognize this fact and, if large losses do oc¬ 
cur in the purely electrical parts of the trans¬ 
ducer, steps should be taken to change the di¬ 
electric materials or their geometry in order to 
reduce these losses. 

It is possible to measure the direct impedance 
between any two terminals of the three-ter¬ 
minal network, represented by the two ter¬ 
minals of the transducer and ground, by means 
of the impedance bridge shown in Figure 2. This 
is a Schering bridge modified by the addition of 
a Wagner ground. The function of the Wagner 


ground is to eliminate all capacitances to 
grounded terminals or to grounded shields. 
When the bridge, including the Wagner ground, 
is completely balanced, terminals A and D are 
both at ground potential. Under these condi- 



Figure 2. Circuit of Schering bridge with 
Wagner ground used for measuring the three- 
terminal impedance of crystal transducers. 


tions, the capacitances to ground from terminals 
A and F cancel out and do not influence the in¬ 
dication of the bridge. The capacitance from A 
to ground merely appears as a shunt across the 
detector. The capacitance from F to ground is 
across the arm E-G. This is compensated for in 
the balancing of the Wagner-ground system, 
since the capacitor normally in the E-G arm is 
reduced by an amount sufficient to compensate 
for the F-G capacitor. The dissipations of the 
capacitances to ground are compensated for by 
adjustment of the variable resistor in the E-G 
arm. The procedure for balancing this bridge is 
first to connect the detector between A to D, 
adjust the capacitors D-F and D-H for a bal¬ 
ance, then shift the detector from A to ground 
and readjust the elements from E-G for bal¬ 
ance. The detector is then reconnected across 
A-D and the procedure repeated as often as 
may be necessary to obtain simultaneous bal¬ 
ance of both the bridge and the Wagner 
ground. 

In measuring the impedance of a transducer 
which has a nonmetallic backing plate, or none 
at all, the direct capacitance of the crystal motor 
may be measured by connecting the case of the 
transducer to the ground terminal G of the 
bridge. The two leads from the transducer 
should be connected to terminals A and F. The 
direct capacitance from either lead to case may 
be measured by grounding the other lead. 

If the transducer has a metallic backing plate, 

















ELECTRICAL MEASUREMENTS 


361 


a lead must be brought out from the backing 
plate in order to make complete measurements 
on the total six capacitances to be found in the 
circuit. These are the capacitance of the crys¬ 
tals, the capacitances of the leads to ground and 
to the backing plate, and the capacitance of the 
backing plate to ground. Each of these six ca¬ 
pacitances and their corresponding dissipation 
may be measured by grounding the two ter¬ 
minals not being used to point G on the bridge. 
A series of six measurements will thus give the 
values of all components of the circuit. 

The capacitances from either lead to the case 
are very small and may be quite difficult to 
measure. All of the transducers developed by 
UCDWR have displayed such small capacitances 
to the case that they have been considered com¬ 
pletely negligible. The capacitance from either 
lead to the backing plate and from the backing 
plate to the case may not be negligible. Further¬ 
more, the dissipation of these capacitors may be 
large enough to cause an appreciable power loss 
and the consequent low efficiency of the trans¬ 
ducer. 

One factor to be considered in any measure¬ 
ment of the impedance of a transducer, espe¬ 
cially when it is operating in water, is the cable 
to which it is connected. Because such measure¬ 
ments must be made at the end of this cable the 
effect of the cable must be taken into account. 
The method of handling this situation in the 
case of direct capacitance measurements is to 
connect the shield of the cable to the ground of 
the bridge and measure the direct impedance be¬ 
tween the two conductors with the transducer 
disconnected. The measurement is then repeated 
with the transducer connected. These final 
measurements may be corrected by considering 
the direct capacitance of the cable to be in paral¬ 
lel with that of the transducer. The cable usu¬ 
ally used with transducers developed at 
UCDWR is identified as Simplex No. 9061. The 
direct capacitance of this cable is approximately 
5.0 pqf per ft. This small capacitance is rarely 
an important factor except when dealing with 
very high impedance transducers. 

As previously mentioned, it is sometimes de¬ 
sirable to measure the overall two-terminal im¬ 
pedance of the transducer as it will be seen by 
the power amplifier, regardless of the arrange¬ 


ment of the internal capacitances. This meas¬ 
urement may be made using the balanced-to- 
ground arrangement, or it may be made with 
one side grounded. The one-side grounded meas¬ 
urement may be made with an ordinary imped¬ 
ance bridge, or the Sobering bridge may be 
used. The more common arrangement, however, 
is to operate the transducer balanced to ground 
since the cable capacitances are thereby mini¬ 
mized and stray fields resulting from the cable 
current are kept at a low value. The calibration 
station at UCDWR employs the balanced sys¬ 
tem wherever possible and, to interpret their 
data, it is necessary to know the apparent im¬ 
pedance of the transducer as seen by a balanced 
line. For this purpose, a bridge of the hybrid- 
coil type has been devised which is similar to 
the Western Electric No. 4A, but which em¬ 
ploys Western Electric Type-146A transform¬ 
ers. This hybrid-coil bridge is locally referred 
to as the “hy-bridge” and has proved very satis¬ 
factory. It is easily operated, requiring only one 
balance adjustment as compared to two in the 
case of the Schering-Wagner bridge, and the 
measurements are directly applicable to the 
calibration data taken on the sound field of the 
transducer. The circuit for this bridge is shown 
in Figure 3. The bridge is symmetrical at both 



Figure 3. Circuit of the hybrid-coil bridge 
used for measuring the complex impedance of 
crystal transducers. 


ends. A variable standard is connected at one 
end and the unknown impedance at the other. 
There is unity ratio between the two ends so 
that the unknown impedance may be read di¬ 
rectly on the variable standards, providing that 
the transformers making up the bridge are well 




























362 


RESEARCH TECHNIQUES AND APPARATUS 


balanced. The two transformers in the main 
loop must be of the very best quality, but the 
transformer leading to the detector is less 
critical. 

A high-gain tuned amplifier is used as the 
detector with both the hybrid and Sobering 
bridges. This circuit contains one or more high- 
Q coils and is equipped with variable capacitors 
so that it may be tuned to any frequency be¬ 
tween 500 c and 500 kc. Suitable selectivity is 
obtained over the entire range. 

The output of this tuned amplifier, or detec¬ 
tor, is connected to the vertical deflection plates 
of a cathode-ray oscillograph [CRO]. The hori¬ 
zontal plates of this oscillograph are connected 
directly to the bridge oscillator so that a Lissa- 
jous figure results. At balance, in the absence of 
harmonics, the Lissajous figure reduces to a 
horizontal straight line. This method of detec¬ 
tion has the advantage that it gives an indica¬ 
tion of both the magnitude and the phase of the 
off-balance voltage with respect to that of the 
oscillator. If compensation is made for the 


dication of which component, in the standards, 
resistive or reactive, needs adjustment. If the 
net phase shift through the system happens to 
be an odd multiple of 90 degrees instead of an 
even multiple, the effects of resistance and ca¬ 
pacitance will be reversed, and the resistive 
changes will rotate the ellipse while the capaci¬ 
tive changes will alter the minor axis. 


PROBE MICROPHONE 

For many experimental purposes, an exceed¬ 
ingly small or point hydrophone that can be 
used either as a contact pickup or as a nondirec- 
tional pickup in a free sound field is a valuable 
tool. Because of its small size, acoustic measure¬ 
ments can be made with a minimum disturb¬ 
ance to the body under measurement. It can also 
follow very short pulses because of its neces¬ 
sarily high resonant frequency. 

The most successful design developed is 
shown in Figure 4. Essentially the hydrophone 



Figure 4. Cutaway of the probe hydrophone. 


phase shift through the system (which is con¬ 
veniently accomplished by a very slight de¬ 
tuning of the detector), then resistance changes 
will alter the axes of the ellipsoidal Lissajous 
figure while capacitive changes will rotate the 
ellipse about its center. This is especially con¬ 
venient if the initial condition happens to be far 
from balance, since it gives the operator an in¬ 


is just two ADP crystals (0.4x0.125x0.06 in.), 
inertia driven, with a metal case and end cap 
serving only for support and electrostatic shield¬ 
ing. Referring to the figure: (1) is the crystal 
motor; (2) is the end cap (0.2x0.2x0.1 in.), 
drawn from 0.001-in. silver foil; (3) is a foam 
rubber side support, serving to center the motor 
and to acoustically insulate it from the case; 
































PROBE MICROPHONE 


363 


(4) is the inside foil which is used as the high- 
potential lead; (5) is the outside electrode foil, 
curled around the foam rubber so as to contact 
the brass case which is at ground potential; (6) 
is a micarta spacer which transmits any me¬ 
chanical thrust on the motor to the end of the 
cable thus relieving the delicate end cap; (7) is 
the external grounded shield; (8) is the interior 
guard shield, driven at the same voltage and 
phase as the central high-potential lead with a 
cathode follower; (9) is the central high-poten¬ 
tial lead that connects to the grid of the cathode 
follower. 

The inside guard shield is used because of the 
exceedingly low capacity of the crystal motor 
(about 10 ppf). The capacity of low-capacity 
cable is usually at least 20 ppf per ft so that a 
cable of 4- or 5-ft length would reduce seriously 
the already small signal from the crystal. If the 
guard shield surrounding the high-potential 
lead is driven with the same voltage and in 
phase with the lead, no current will flow from 
shield to lead, and the effective capacitance be¬ 
tween lead and ground in the cable will be zero. 
This action is accomplished with a cathode- 
follower circuit shown in Figure 5. If a long 



in Figure 6. The voltage measured is that ap¬ 
pearing across the cathode-follower terminals. 
For use as a contact pickup, the input imped- 



Figure 6. Frequency response of probe hydro¬ 
phone in water (volts/dyne/cm^). 



Figure 7. Directivity pattern of probe hydro¬ 
phone (at 125 kc) in water, in a plane containing 
the axis. 


Figure 5. Cathode-follower guard circuit for 
use with probe hydrophone. 

cable is necessary, this simple circuit must be 
replaced by several stages of a 100 per cent 
feedback amplifier that can furnish enough 
power to drive the guard. 

The directivity pattern of the probe taken in 
a plane containing the center line of the motor 
is shown in Figure 7, and its sensitivity cali¬ 
bration in volts per dyne per sq cm pressure 


ance was measured to be 4,700 mechanical 
ohms, which is low compared to 1.5 X 10^ the 
specific acoustic resistance of water. Thus, for 
contact measurements, it looks to the surface 
under measurement more like air than water 
loading. Absolute calibrations of the unit as a 
contact pickup are not measured. However, it is 
used principally for relative measurements, so 
that its linearity is of chief importance. This 
must be checked over the range used at the be- 


































































364 


RESEARCH TECHNIQUES AND APPARATUS 


ginning of every measurement, because these 
ranges are of the order of 30 db. 

The complete vibration pattern of the surface 
of a transducer at a single frequency is ob¬ 
tained by scanning the probe over the surface, 
probe being coupled to the motor surface by a 
thin oil film (about 0.005 in. thick). A scanning 
machine that moves the transducer or motor 
under the probe is shown in Figure 8. This ma- 


9 3 direct-reading phase meter 

Many methods of measuring phase differ¬ 
ences between two circuits have been advanced; 
they can be divided generally in two classes, 
those which measure changes in magnitude of a 
resulting voltage with change of phase,^ and 
those that measure directly the time difference 
between a certain point of a cycle of one circuit 



Figure 8. Jig for scanning transducer motors in air using a probe contact microphone. 


chine keeps the probe spacing from the motor 
constant, and enables the operator to quickly 
scan a motor point by point. It can also be made 
automatic. Scanning under oil is shown in Fig¬ 
ure 9. The results of a motor scanned both in air 
and oil are shown in Figure 21, Section 3.6. The 
phases indicated in that figure were measured 
by a phase meter described in Section 9.5. The 
oil bath for loading the motor must be free of 
standing waves that come from boundary re¬ 
flections. This is accomplished by enclosing the 
oil in QC rubber case which is then immersed in 
a water tank that has acoustically absorbing 
walls, as illustrated in Figure 9. 


and the corresponding point on a cycle of the 
other circuit.2 The first method is essentially a 
point by point measurement, while the second 
can be made direct reading, and even recording. 
The present phase meter is based upon the sec¬ 
ond method and indicates phase differences di¬ 
rectly with a d-c milliammeter. The overall 
operation is illustrated in Figure 10. Referring 
to the figure, each sine wave signal is squared 
and differentiated into successive positive and 
negative pips. These pips are used to trigger a 
flip-flop circuit (Figure 11) that works only on 
negative pips, each channel being fed into op¬ 
posite sides of the circuit. The effect is that one 








DIRECT-READING PHASE METER 


365 


channel turns a tube on and the other channel 
turns the same tube off. The current through 
the tube thus flows only during the interval of 


to use a flip-flop circuit that works with nega¬ 
tive pulses instead of positive because in the 
latter type the square wave generated always 



Figure 9. Cutaway, showing method of scanning transducer motor in oil using probe contact hydrophone. 


time representing the difference in phase be¬ 
tween the two circuits. These pulses are aver¬ 
aged in time with a d-c milliammeter whose 
readings are then directly proportional to phase 
differences in channels A and B. It is necessary 



Figure 10. Wave form diagram showing over¬ 
all operation of phase meter. 


has a small initial pip superimposed at the be¬ 
ginning of each wave, and this perturbation 
introduces considerable error in a measurement 



Figure 11. Flip-flop circuit employed in the 
phase meter which is sensitive to only negative 
impulses. 













































































































366 


RESEARCH TECHNIQUES AND APPARATUS 


that depends upon short time intervals of con¬ 
stant current for its linearity. Also, the use of 
negative pips allows phase readings over a 
range of 0 to 360 degrees. 

The operation of the circuit of Figure 11 is 
as follows. Assume tube B is conducting and A 
is shut off. The suppressor of A is about 50 volts 
less than the cathode so that a positive impulse 
at the grid of A will not disturb the plate cur¬ 
rent of the tube. Only by raising its suppressor 
can this tube be rendered conducting, and this 
happens when tube B is shut off. A negative 


there will be beat frequencies of the form 

E[ = sin [(coi — oi 2 )T + </>i + 0], 

E 2 = sin [(oji A- 0 : 2 ) T -j- <p 2 + 6], 

and the phase difference between these signals 
is still ((/>! — c^o). 

Operating the flip-flop circuit at constant fre¬ 
quency gives two additional advantages in that 
phase variations with frequency in the am¬ 
plifiers and in the standard phase shifter are 
eliminated. 

Referring to Figures 12 and 13, the complete 



Figure 12. Block diagram of the phase meter. 


pulse on the control grid of B will stop the flow 
of electrons through it momentarily whence its 
plate voltage rises, thus raising the suppressor 
of A and rendering it conducting. The plate of 
A now is depressed which depresses the grid of 
B thus keeping it shut off until a negative pip 
at the grid of A shuts it off thus allowing B to 
conduct again. At frequencies up to 10 kc the 
wave through tube B is perfectly square as indi¬ 
cated in Figure 10, but at higher frequencies 
the corners begin to round a little, which intro¬ 
duces errors in phase readings close to zero or 
360 degrees. To extend this frequency range, the 
input signals in each channel are beat down to 
2.2 kc with a common oscillator. This transfor¬ 
mation does not change the phase relation be¬ 
tween the channels, for if two signals of the 
same frequency but different phases 

El = sin (coiT -f- 0 ), 

E 2 = sin (coiT + </> 2 ), 

are beat with a second frequency 

E^ = sin {W 2 T A G), 


operation of the instrument is as follows: Each 
signal is fed into its respective channel through 
a high impedance input cathode follower input 
tube. The cathode follower then feeds into a 
limiting amplifier or automatic volume control 
[AVC] system which compensates for wide 
ranges in input-signal level. The resulting 
nearly constant signals are then mixed with a 
frequency from a tuned oscillator which 
changes their frequencies to 2.2 kc. The lower- 
frequency signals are then put through 2.2 kc 
tuned amplifiers to second-volume limiting am¬ 
plifiers. The output of these second amplifiers 
is constant with about 30 db variation in the 
voltage of the input signal at the cathode fol¬ 
lower. At this point channel A is fed directly 
into a square-wave generator, while B is fed 
through a standard phase shifter to a square- 
wave generator. This phase shifter allows phase 
compensation for the preceeding stages, accu¬ 
rate measurement of phase differences, and a 
means of adjusting for zero phase shift in the 
final circuits. From the square-wave generators 













































































































































































































































































































































































































































































































































DIRECT-READING PHASE METER 


367 


the signals are fed through differentiating 
filters to the final flip-flop measuring circuit. 

The accuracy of the phase meter depends 
upon the linearity of the milliammeter, wave 
form of the flip-flop circuit square wave, sharp- 



Figure 14. Five-point calibration of phase 
meter. 


ness of the triggering pips to the flip-flop cir¬ 
cuit, and readability of the indicating meter. In 
principle, all the factors can be held constant, 
so that a very accurate meter is theoretically 



Figure 15. Calibration of phase meter by the 
two-oscillator method. 


possible. In the calibration of the meter, three 
methods can be used and checked against each 
other. 

1. Obviously, one method would simply com¬ 
pare the meter reading with a standard phase 


shifter. At different frequencies, however, the 
calibration of such a shifter is in doubt, so that 
this method is restricted to the standard shifter 
incorporated in the meter that works at 2.2 kc. 

2. The second method is as follows: Refer¬ 
ring again to Figure 11, when tube B is con¬ 
ducting continuously, the corresponding phase 
shift should be 360 degrees and when it is shut 
off the shift should be 0 degrees. If the meter 
current is plotted for these two points. Figure 
14, and connected with a straight line, the meter 



Figure 16. Large auxiliary indicator used with 
the phase meter. 

reading for 180 degrees should lie on this line at 
half the distance between the zero and 360 de¬ 
grees. A 180-degree phase shift in a high-fre¬ 
quency input signal is easily obtained from the 
secondary of a transformer by reversing the 
terminals. Actually, the 180-degree point lies 
exactly upon the curve as shown in Figure 14. 
Phase shifts of 90 and 270 degrees can be ob¬ 
tained from calibrated RC elements and, when 
used with the meter, fall exactly on the calibra¬ 
tion line. 

3. The third method requires two stable 
oscillators, one at each high-frequency input 
terminal, tuned at slightly different frequen¬ 
cies. With two such incommensurate frequen¬ 
cies, the phase difference in the two channels 

















368 


RESEARCH TECHNIQUES AND APPARATUS 


varies linearly with the time (the linearity de¬ 
pending upon the stability of the oscillator). 
The meter reading should then vary linearly 
with time and experiments show that a very 



Figure 17. Panel view of phase meter. 


good linear variation with time is possible with 
this test. This calibration is illustrated in Fig¬ 
ure 15. 

Photographs of the phase meter, omitting the 
power supply, are shown in Figures 16, 17, and 
18. Figure 16 is an auxiliary indicating meter 



Figure 18. Chassis view of phase meter. 


of a large type to give better scale readability. 
Figure 17 illustrates the controls of the phase 
meter. The operation is as follows; The oscilla¬ 
tor dial on the right is first tuned to the input 
frequencies. Then the gain in each channel is 
turned to zero so that the zero- and the 360-de¬ 
gree currents in the indicating milliammeter 
can be adjusted with the knobs under the meter. 
These knobs operate shunting potentiometers 
across the meter as illustrated in Figure 13. 
Finally the gains in each channel are set so that 


the two meters on each side of the central meter 
which read input signal to the square-wave gen¬ 
erators read about half scale. It is necessary to 
keep the input signals strong enough to give 
good square waves but not so strong as to over¬ 
load any of the amplifiers. Because of the two 
volume limiting controls, the levels of the input 
signals can vary through considerable range 
(0.023 to 5 v). Any phase-shift zero can now be 
set with the phase shifter, and the meter is 
ready to read or record phase differences. Zero 
phase can be set by paralleling the two input 
channels, and setting the phase indicating meter 
to zero with the standard phase shifter. Figure 
18 illustrates the chassis of the meter. 


^ ^ PULSE MODULATOR 

For direct investigations of the behavior of a 
circuit or transducer when being pulsed at short 
intervals with a given frequency, it is neces¬ 
sary to have a signal generator that will gener¬ 
ate such pulses of controllable durations and 
repetition rates without transient build-up or 



Figure 19. Block diagram of the pulse modu¬ 
lator. 


decay perturbations. In addition it is desirable 
that the carrier frequency in the pulse be syn¬ 
chronized with the repetition rate of the pulse 
so that it will appear stationary when viewed 
on an oscilloscope screen. 

All these controls are embodied in the present 
pulse modulator. Referring to the block diagram 
of Figure 19, the overall operation of the cir¬ 
cuit is as follows: A repetition rate is first cre¬ 
ated by the pulse initiator, which is a saw-tooth 
oscillator synchronized to the carrier frequency 
in such a way that the sharp rise, or firing time, 
occurs only at a definite point on the positive 
swing of a carrier cycle. This rate may be 




















PULSE MODULATOR 


369 



Figure 20. Schematic circuit diagram of the pulse modulator. 














































































































































































370 


RESEARCH TECHNIQUES AND APPARATUS 


varied, but is always synchronized with some 
cycle of the carrier. This sawtooth signal is 
sharpened by a differentiating circuit, to allow 
for very short pulses, and used to trigger a 
“one-shot” multivibrator. The duration time of 
this multivibrator is determined by its resist¬ 
ance and capacitance, and is the duration of the 
final pulse. The multivibrator’s square wave 
output is then amplified and used with a bal¬ 
anced modulator to shape a pulse of the carrier 



Figure 21. Panel view of the pulse modulator. 

signal having the desired duration and repeti¬ 
tion rate. 

The complete schematic circuit diagram ap¬ 
pears in Figure 20. A sawtooth oscillator utiliz¬ 
ing a 2050 tube is used for the pulse initiator be¬ 
cause it synchronizes with the carrier over a 
much wider band of frequencies than do con¬ 
trolled multivibrators. The 2050 tube, whose 
screen is controlled by the carrier voltage, is 
used to discharge an RC charging circuit. The 
RC time constant controls the repetition rate 
generally, but the carrier voltage at the 2050 
screen determines the precise time of discharge, 
thus synchronizing the square-wave modulating 
voltage with the carrier itself. The repetition 
rate is adjusted in large steps by capacity varia¬ 
tions and in fine steps by a variable resistor. 
The one-shot multivibrator is likewise time con¬ 
trolled in large steps by a step-switch to capaci¬ 
tors, and in fine by a variable resistor. The 
usual type of multivibrator is inadequate to 
monitor very short pulse lengths in that it 
ceases to have a one-shot action and operates as 
a continuous multivibrator without control. 
(The cause of this instability at high frequen¬ 
cies lies in the high-impedance couplings be¬ 
tween the grids and plates; low-impedance 


loading upsets the frequency of the circuit.) A 
modified form of the conventional one-shot mul¬ 
tivibrator using cathode output shown in the 
diagram is necessary if short pulses are to be 
produced. As the output of the multivibrator is 
low, it must be amplified, then fed to the modu¬ 
lator at low impedance. This is done with the 
amplifier and cathode-follower tubes following 
the multivibrator. The cathode-follower load 
resistor is common to the cathode resistor of the 
balanced modulator and keeps the modulator 
biased to cutoff except when there is a signal on 
the cathode follower. The carrier signal is fed 
push-pull through the modulator, and is 
chopped into packets determined by the cathode 
modulating square-wave voltage. It is necessary 
to connect the two screens in the modulator 
through a potentiometer to a source of very 
constant voltage as shown. This adjustment 
allows a very accurate balancing of the output 
signal to be made, which is necessary in order 
that the zero-voltage axis of the pulse be a 
straight line. 

Photographs of the pulse modulator appear in 
Figures 21 and 22. Figure 21 illustrates the 
panel controls and Figure 22 illustrates the 
chassis. The synchronizing knob adjusts the 
input voltage to the screen of the 2050 tube that 



Figure 22. Chassis view of the pulse modulator. 

acts as the pulse initiator, which adjustment is 
not critical. Two knobs are used on the repeti¬ 
tion-rate control and two on the pulse-length 
control. In each case, the lower knob operates a 
capacitor step switch and the upper know op- 






USE OF 45-DEGREE X-CUT RS AS A RESEARCH TOOL 


371 


erates a 1-megohm potentiometer. The balance 
control operates the potentiometer between the 
screens of the balanced modulator tubes. This 
control straightens out the horizontal axis of 
the pulse. An extra pair of terminals, not 
marked on the chassis, connects directly to the 
output of the differentiating circuit to give a 
synchronizing pip for use in synchronizing a 
single-sweep oscilloscope. 


USE OF 45-DEGREE X-CUT RS AS A 
RESEARCH TOOL 

It is well known that X-cut RS exhibits 
marked temperature dependence at tempera¬ 
tures commonly encountered in the oceans. This 
temperature dependence causes large changes in 
the dielectric constant K, electromechanical 
coupling coefficient k, velocity of sound in the 
crystal V, characteristic impedance Zq, and 
transformation ratio </>. For this reason 45-de¬ 
gree X-cut RS crystals are no longer used ex¬ 
tensively in newly designed equipment. How¬ 
ever, as will be discussed in the following text, 
there still remains an important possible use for 
these crystals in research, the results of which 
would allow the design of improved transducers 
using other more stable crystals. 


Radiation Problems 

An important class of transducer is that 
which has at least one dimension comparable to 
a wavelength. Our present theories of radiation 
all make use of an approximation that the 
transducer is either very large or very small 
compared with a wavelength; the inadequacy 
of this theory has been pointed out several 
times in this book (Sections 4.4.2 and 6.9.3). 
It is not now possible to calculate the directivity 
pattern, the point radiation impedance, or 
even the average radiation impedance of such 
transducers. The theoretical problems are very 
great, and no solutions are now in sight. The 
theoretical treatment would be aided greatly 
by experimental data for several typical radi¬ 
ators. These experimental data on directivity 
patterns are easily obtained, and much is now 


available. However, there is available no infor¬ 
mation concerning radiation impedance. 

The most natural way to obtain radiation- 
impedance information is from measurements 
made at the electric terminals of an efficient 
transducer. However, in both 45-degree Y-cut 
RS and 45-degree Z-cut ADP the electrome¬ 
chanical coupling coefficient is 0.3; consequently 
one cannot “see” the mechanical branch very 
well because of the low shunt impedance of Co. 
Even at resonance, the best available measure¬ 
ments of complex impedance allow very poor 
accuracy in computing the impedance of the 
mechanical branch. This situation would be im¬ 
proved by using a crystal having larger k. 


’ X-Cut RS 

According to Froman^ the apparent dielectric 
constant is not only temperature dependent but 
also very strongly field dependent, and neither 



5 10 15 20 25 30 35 


*C 

Figure 23. Electromechanical coupling coeffi¬ 
cient k of 45° X-cut Rochelle salt crystals as a 
function of temperature. 

dependence is monotonic at any value of the 
other variable. However, at low fields Froman 
observes the same temperature dependence as 
Mason,'* a smooth single-peaked curve. It is 
thought that at least within the Mason approx¬ 
imation at low fields the entire temperature 
dependence can be ascribed to K, the other 
quantities varying only as they are functions 
of K. If so, and if the available data on K versus 
temperature at low field are correct, we can 

















372 


RESEARCH TECHNIQUES AND APPARATUS 


compute the behavior of 45-degree X-cut RS at 
any temperature and low field. In particular, we 
may compute /c as a function of temperature; 
using Mason’s data we obtain the curve shown 
in Figure 23. 

The band width of a transducer, and also the 



5 10 IS 20 25 30 36 

«C 


k- 

Figure 24. Figure of merit of 45° X-cut 

Rochelle salt crystals as a function of tempera¬ 
ture. 


sensitivity of measurements of the mechanical 
branch from the electric terminals, is governed 
primarily by the function 

i - ^2- 

Using the data in Figure 23, this function is 
shown in Figure 24. 

It is seen from Figure 23 that at approxi¬ 
mately 24 C, X-cut RS is an enormous improve¬ 
ment over Y-cut RS or Z-cut ADP. 


""" Technique 

The large values of k are found only over a 
very restricted temperature range. This is a 
temperature above that usually encountered at 
calibration stations, but not one difficult to 
obtain. It is suggested that test transducers be 
built using 45-degree X-cut RS. These could 
have radiating faces whose dimensions are of 


interest. If these units and a large volume of 
water around them were maintained at very 
constant temperature near 24 C it should be 
possible to measure complex electrical imped¬ 
ances which allow calculation of the average 
radiation impedance as a function of frequency. 
These measurements would be done at low field 
where the assumed K should be correct. Under 
such conditions it is not uncommon that the me¬ 
chanical arm be coupled so strongly that the 
transducer’s series reactance go inductive above 
resonance. 


Difficulties 

Many difficulties present themselves, but none 
appears insurmountable, and the end appears 
to justify elaborate means of overcoming them. 

The purely mechanical difficulties of main¬ 
taining a sufficiently large body of water differ¬ 
ent from ambient temperature are engineering 
problems whose solutions are straightforward. 
Some trouble may arise from reflections at the 
temperature interface; for this reason the inter¬ 
face should be as remote as possible. 

Before this method can succeed the data on 
dielectric constant versus temperature and field 
must be checked and perhaps taken to greater 
accuracy. It is also necessary to show that all 
of the temperature dependence can be embodied 
in K. 

Having computed the series impedance of the 
mechanical branch, it is then necessary to know 
that part caused by the crystal so it can be sub¬ 
tracted to determine the radiation impedance. 
For this it may be necessary to determine the 
constants in Camp’s equivalent circuit. 

The X-cut RS is not only temperature de¬ 
pendent, but it also shows marked nonlinearity 
and hysteresis. The nonlinearity may arise from 
the field dependence and may not be serious at 
low fields. The hysteresis introduces a resistive 
component, some of which may appear in the 
electrical branch. This resistance would have to 
be determined. 

It is necessary to use highly efficient trans¬ 
ducers and to know the values of the loss re¬ 
sistances. This may be difficult, but should be 
possible if gas-filled inertia drive is used. This 






















ELECTRIC NETWORK SIMULATOR OF THE COMPLETE TRANSDUCER 


373 


drive has the further advantage of lower elec¬ 
trical Q 


ELECTRIC NETWORK SIMULATOR 
OF THE COMPLETE TRANSDUCER USING 
CONSTANT L, C, AND R ELEMENTS 

The impedances appearing in the equivalent 
circuit for a piezoelectric crystal being tran¬ 
scendental functions, constant circuit elements 
of inductance, capacitance, and resistance can 



Figure 25. Equivalent circuit of a piezoelectric 
crystal (Mason^). 

replace them directly at only one frequency in 
an electric simulating circuit. These functions 
are tangents, cotangents, and cosecants in non- 
dissipative systems and are the hyperbolic func¬ 
tions in dissipative systems. The problem of 
calculating a transducer’s frequency response 
from an equivalent circuit of transcendental 
elements is very laborious, and it is more prac¬ 
tical, and far more time saving, to build an 
equivalent circuit from constant L, C, and R ele¬ 
ments, and measure its response with frequency. 
Transcendental impedance approximations 
using lumped circuit constants are derived in 
the following manner: 

The dissipationless equivalent circuit of a 
piezoelectric crystal of length I and area A is^*' 
shown in Figure 25, 

where Z = pcA; A is cross-sectional area, 

Co = capacity of blocked crystal, 

Fi and F 2 = the longitudinal forces on the 
ends of the crystal, 

(t> = piezoelectric-mechanical coupling 
coefficient. 

The impedance elements in the T network can 
be approximated by considering the crystal bar 


to be cut into short equal lengths 1. Each length 
is represented by a T section in which the argu¬ 
ments kl are small. The equivalent circuit of 
the whole bar will then be a chain of these ele¬ 
ments. The following approximations under 
these assumptions are valid: 


tan X = X, 
1 

CSC X = -' 
X 


( 1 ) 


It can be shown that four sections of lengths 1/4 
are adequate to give excellent accuracy, so that 
the following approximations are valid: 


, kl 

kl 

tang 

~ 8’ 

kl 

kl 

sec—r 

4 

4 • 


In the complete transducer of n crystals the 
electrical impedances are 


and 


where 


pcA 

n<j>^ 



Al 


pcA kl 1 

—^ CSC -j- = —Tf , 

n<l>~ 4 n(i)H 

Oi —r-r 



X ■ 


From these approximations we see that con¬ 
stant inductances and capacitances 


T = 

... 

r - 

^ “ 4pc^A’ 

can be used in the T branches of the bar ele¬ 
ments. Each coil and condenser shown in Fig¬ 
ure 26 have the values as given above. 


o-qnnrvTnnp-'Tnrir^llTnnr'^^ 

t-1' 


Figure 26. Four-section transmission line of 
constant L and C elements which is an approxi¬ 
mation of a T network of transcendental ele¬ 
ments. 


With these four elements the short-circuited 
resonance (or free-free analog) occurs at kl = 
3.07 which is to be compared to kl = it, the 
















374 


RESEARCH TECHNIQUES AND APPARATUS 


short-circuited T network of three transcen¬ 
dental impedances. The complete transducer 
equivalent circuit using L and C constants for 
n crystals can be well represented up to the first 
free-free resonance by Figure 27. 

If the transducer is inertia driven, the back¬ 
ing plate impedance is zero, or a short circuit. 
The radiation impedance is assumed to be pure 


Each term of this sum represents a parallel LC 
circuit whose impedance is given by 



and whose antiresonant frequency is = 
{2n — l)/o and whose L and C constants are 



Figure 27. Approximate equivalent circuit of 
the complete crystal transducer. 


resistance which in water is pcA/c^-n in cgs esu 
units. The backing plate being a transmission 
line terminated with zero impedance, its im¬ 
pedance could be represented by the circuit of 
Figure 26, short-circuited at the terminal end. 
The impedance of this circuit is, however, a 
single tangent function of the argument kl^ 
(^j, thickness of the plate) and can be repre¬ 
sented by only three or four L and C elements, 
using a partial fraction expansion.^ 


tan X = 


n = cD 

z 


8x 


{2n - 1)V^ - 4x2’ 


or Z = jZ^tanlk 




8kl 


{2n - 1)2^2 - 4(y^/)2- 


(3) 


This expansion can be rewritten to represent 
an infinite sum of parallel LC circuits whose 
values are given in terms of Zo and the quarter- 
wave frequency /o of the plate. At this fre¬ 
quency kj, = 7r/2 SO that kl = = (7r/2)- 

(///o)j where / is any frequency up to /o. 

Thus 


^ 2Zo ^ 8M 
/o(2n - 1)2x2 x2(2n - 1)2’ 

r _ 1 _ 1 _ L 

(2x /(^HZq 8/oZo 2pcA* 

where M is the mass of a section of area A in 
the plate. Equations (5) and (6) show that the 
capacitances do not vary with n, but the in- 




Figure 28. Approximate equivalent circuit of a 
backing resonator, 

ductances decrease in ratios 1: 1/9; 1/25; 1/49; 
etc., so that ultimately the capacitive reactance 
may be neglected without appreciable error. 
Since f < fo the denominators in equation (4) 
are appreciably one after the first term and 
equation (4) becomes 

2jo:Zo 9 7 

In terms of total mass of the section equation 
(7) can be written 



(5) 

( 6 ) 


y _ j(ji {8 TV-) M X . 8M 
^ “ 1 - (P//5) ^ - 1)^’ 


( 8 ) 


j\i^{f/h)ZM{2n-X)V 

1 - [fyfliZn - 1)’] ’ 

which may be written 

. (co2Zo)//o(2/l - 1)2x 2 
^ 1 - [/V/i(2n - 1)T 


which is the equation of one parallel LC circuit 
whose resonance is /o in series with a string of 
inductances of decreasing magnitude. The in¬ 
ductance of the first term is close to S/ji^ = 0.813 
of the total mass since ///o < 1 which leaves 
only 0.187 of the mass for the series elements. 
This approximation is exact at 0 and /o, and is 






































ELECTRIC NETWORK SIMULATOR OF THE COMPLETE TRANSDUCER 


375 


off less than 0.3 per cent at /o/2. The resultant 
equivalent circuit for a backing plate is in Fig¬ 
ure 28, 


where h = —, M, L 2 = 0.181 M, 

TT- 


, ^ 1 
' 32/,W 


The equivalent circuit for a complete trans¬ 
ducer with a backing plate and radiating into 



Figure 29. Approximate equivalent circuit of a 
complete transducer using a backing plate and 
radiating into water. 

water can now be drawn. Using the network 
shown in Figure 29, the following equations can 
be set down: 


Co = 


KA 


Lb = 

Cb = n^- 


An-irlc 
1 8pAIb 

^ TT^ ’ 

Ib 


2pcA’ 

pcA 


R = 


n<i>- 


It now remains to be shown that the LC 
and R elements are practically realizable for a 
crystal. The resonant frequency of an LC ele¬ 
ment of Figure 26 can be calculated from equa¬ 
tion (2) as 





O) 


2 

C) 


where = (^/2) {C/IJ, the frequency of the 
first free-free resonance of the crystal bar, 
which occurs at kl = .-t/2. Thus each LC element 
must resonate at 8\/2/k = 3.59, instead of 4 
times the resonance of a crystal blocked on one 


end, which is certainly practical. As it is diffi¬ 
cult to vary inductances and easy to vary ca¬ 
pacitances, fixed inductors of a convenient size 
may be chosen, and different capacitors used to 
represent different transducers. The range of 
types is limited, however, by equation (2). Re¬ 
ferring to this: 


InA =-^andC 

p<ir a 4pc- 


Thus if L is fixed there is a definite relation 
InA = constant, which means that only two of 
the parameters n and A are independent. A con¬ 
venient value of inductance equaling 3 X 10“^ h 
resonates with a capacitance c equaling 0.013 pf 
at 25.5 X 10^ c. The circuit of Figure 26 using 
these values will represent a short-circuited line 
or tangent function whose characteristic imped¬ 
ance Zo = 680 ohms whose first resonance occurs 
at 7.08 X 10^ c. 


For Zo = —= 680 ohms, 

n4r ’ 

_ pAc I 
8n<p^ ~ ncj)- 8c 

For the tangent function 


_ 

~ ~C~ ~ 2’ 


Ilf 25.5 X 10' 
C 4// 3.6 


= 7.08 X 10', 


Zo = 32/rL = 680 ohms. 
A plot of the function 


Zo 680 tan ^ ^ j 

together with the measured impedance of the 
short-circuited transmission line is given in 
Figure 30. The agreement between the two is 
excellent up to 6.5 kc. Beyond that, the resist¬ 
ances in the inductances become effective, and 
throw the experimental values off somewhat. 

If the transmission line be terminated in an 
open circuit, the input impedance will approxi¬ 
mate a cotangent function. Figure 31 illustrates 
the experimental input impedance of this cir¬ 
cuit together with the mathematical cotangent 
curve. The agreement between the measured 





















376 


RESEARCH TECHNIQUES AND APPARATUS 


8000 

6000 


















Z=ZQTan 

Zo =68 
Ko = 21 

KoL 

0 



I 










TFq 

c 












SOLID LINE : EXPERIMENTAL 
THEORETICAL POINTS! 0 




1 

oo; 

KTHT} 


iTgr 




•Tnn 

rvi 

✓inn 


4000 

y 2000 

z 

< 

o 

UJ 

Q. 

z 

H 

3 

Q. 

Z 

1 

to 

X -2000 

-40 0 0 

-6000 

- 8000 








'1 



^ i 

k^ 

k _ t 

._r' 









J 

>wv 

—<y)- 














F^= 25 

c -Tn 

5KC 

Q Kr 


J 











ro-'* 


























8 

9 

10 

11 

12 

13 


1 

2 

3 

4 

5 

6 

7 

KC 











































'- 












































D 












































/ 









Figure 30. Experimental performance of the transmission line of Figure 26 connected so as to simulate 
the tangent function. 







































































ELECTRIC NETWORK SIMULATOR OF THE COMPLETE TRANSDUCER 


377 



Figure 31. Experimental performance of the transmission line of Figure 26 connected so as to simulate 
the cotangent function. 















































































378 


RESEARCH TECHNIQUES AND APPARATUS 



Figure 32. Application of the simulator for a backing plate transducer radiating into water. 



WATTS IN 261 -n. RESISTORS 




































































ELECTRIC NETWORK SIMULATOR OF THE COMPLETE TRANSDUCER 


379 


impedances and the tangent function is remark¬ 
able. 

A steel backing plate simulating circuit 
resonating at 7.5 kc can be built of two in¬ 
ductances of 0.014 and 0.056 h, and a capac- 



Figure 33. Application of the simulator for a 
hypothetical transducer, radiating into water, 
in which the back end of the crystals are blocked 
at all frequencies. 

itance of 0.008 ^ipf as shown in Figure 32. 
When the simulating circuit is loaded at one 
end with this circuit, and at the other with a 
resistance of 261 ohms, it represents a trans¬ 


ducer whose crystals resonate at 14.15 kc, ter¬ 
minated at one end in a backing plate whose 
resonance is 7.5 kc, and radiating into water 
the resistance of which is Zo = QcA/n^r, where 
A is the area of a single crystal. Curves of input 
impedance and velocity response at the radiat¬ 
ing end of this circuit are shown in Figure 32. 
The resonant frequency is 7.02 kc which is just 
half that of the free-free crystal. The response 
shows a mechanical Q of about 3. The response 
of a real transducer is not as regular, of course, 
as this curve, because it is constructed of many 
crystals attached to a backing plate that has its 
own flexural modes through glue joints that are 
not uniform. The general overall response of a 
real unit does, however, resemble somewhat the 
response curve of the simulator. See Sections 
4.5 and 4.6. 

Figure 33 illustrates the simulation of a 
transducer whose calculated resonance is 30.5 kc 
when the crystals are blocked at all frequencies. 
Practically, this is not possible to achieve, be¬ 
cause complete blocking can only be done with 
a quarter-wave resonator. However, the simu¬ 
lation represents the real transducer in the 
resonant region. The power dissipated in the 
water-loading resistor is plotted against fre¬ 
quency. As the power scale is logarithmic, the 
response appears to be quite broad. By shorting 
terminals A and B, the same circuit simulates 
the response of an inertia-driven transducer 
which resonates at 61 kc or double the resonant 
frequency shown in Figure 33 with terminals 
A and B open. The general shape of the re¬ 
sponse curves in both cases is theoretically the 
same but with the ordinate scale in a ratio of 
2 to 1. 





























































GLOSSARY 


ADP (Crystal). Ammonium dihydrogen phosphate 
crystal having marked piezoelectric properties. 

Ambient Noise. Noise present in the medium apart 
fi’om target and own ship noise. 

BDI. Bearing Deviation Indicator. 

Bearing Deviation Indicator. A system which utilizes 
the outputs of the halves of a split transducer to 
provide accurate directional indication. 

Bimorph (Crystal), a rigid combination of two 
Rochelle salt crystals designed to give improved 
coupling betv/een the crystal element and low (me¬ 
chanical) impedance devices, such as telephone 
diaphragms and loudspeaker mechanisms. 

BTL. Bell Telephone Laboratories. 

Cavitation. The formation of gas or vapor cavities 
in a liquid, caused by sharp reduction of local pres¬ 
sure. 

Clamped Drive. A condition in which the radiation 
impedance (water) is on one end of the crystal and 
a backing plate is on the other. 

Crystal Transducer. A transducer which utilizes 
piezoelectric crystals, usually Rochelle salt, ADP, 
quartz, or tourmaline. 

CUDWR. Columbia University Division of War Re¬ 
search. 

Cycle-Weld. A commercial cement. 

Dilatation. The increase in volume per unit volume 
of a very small undisplaced region of a substance. 

Directivity Factor. A measure of the directional 
properties of a transducer. It is the ratio between 
the average intensity, or response, over the whole 
sphere surrounding the transducer, and the intensity, 
or response, on the acoustic axis. 

Directivity Index. Directivity factor expressed in 
decibels. 

Dome. A transducer enclosure, usually streamlined, 
used with echo-ranging or listening devices to mini¬ 
mize turbulence and cavitation noises arising from 
the transducer’s passage through the water. 

Eigen Mode. Natural mode of vibration. 

Electret. The electrical analogue of a magnet. 

Eutectic Alloy. An alloy with the lowest melting 
point of all alloys containing the same constituents. 
Upon solidifying all the constituents crystallize 
simultaneously. 

Hydrophone. Underwater microphone. 

Induction Field. The region, immediately surround¬ 
ing a transmitter face, where the inverse square law 
does not hold. 

Inertia Drive. A condition in which the radiation 
impedance (water) is on one end of a crystal and 
zero impedance (air) is on the other. 

Matrix (Crystal). An assemblage of crystals for use 
as a transducer. 

Metastable State. A state, actually unstable, which 
appears stable because of the length of time it per¬ 
sists. 


Motor, Crystal. The crystal assembly and backing 
plate in a transducer. 

Neoprene. Generic name for synthetic rubber made 
by polymerization of 2-chloro-l,3-butadiene. Vul- 
canizates are markedly resistant to oils, greases, 
chemicals, sunlight, ozone, and heat. 

NRL. Naval Research Laboratory. 

Optic Axis (Z-axis). The direction in a doubly re¬ 
fracting crystal along which the ordinary and 
extraordinary light rays pursue the same path with 
the same velocity. 

Orthorhombic System. A system of crystals having 
three unequal axes at right angles to each other. 

Piezoe}lectric Effect. Phenomenon exhibited by cer¬ 
tain crystals in which mechanical compression pro¬ 
duces a potential difference between opposite faces, 
or an applied electric field produces corresponding 
changes in dimensions. 

Ping. Acoustic pulse signal projected by an echo¬ 
ranging transducer. 

Probe Microphone. A very small crystal microphone, 
used to study variations in sound output. 

Reciprocity Principle. The ideal transducer gives 
analogous performance in transmission and recep¬ 
tion; for example, the directivity patterns are the 
same in both cases. 

Reverberation. Sound scattered diffusely back to¬ 
wards the source, principally from the surface or 
bottom and from small scattering sources in the 
medium such as bubbles of air and suspended solid 
matter. 

pc Rubber. A rubber compound with the same pc 
(density times velocity of sound) product as water. 
Also called sound, or sound-water, rubber. 

Rochelle Salt. Potassium sodium tartrate, 
KNaC 4 H 40 o' 4 H 20 , piezolectric crystal used in sonar 
transducers. 

RS Crystal. Rochelle salt crystal. 

Symmetric Drive. A condition in which the radiation 
impedance (water) is on both ends of a crystal. 

Sound (sound-water) Rubber. See pc rubber. 

Tetragonal System. A crystallographic system in 
which all the forms are referred to three axes at 
right angles; two are equal and are taken as the 
“horizontal” axes; the remaining “vertical” axis is 
either longer or shorter than the others. 

Thermoplastic Substance. A substance which be¬ 
comes plastic upon being heated. 

Thermosetting Substance. A substance which, upon 
the application of heat, acquires and retains new 
chemical and physical properties which subsequent 
application of heat does not alter. 

TRANSDUCiai. Any device for converting energy from 
one form to another (electrical, mechanical, or 
acoustical). In sonar, usually combines the functions 
of a hydrophone and a projector. 

UCDWR. University of California Division of War 
Research. 


381 



382 


GLOSSARY 


Varistor. Nonlinear resistance whose value decreases 
with increasing applied voltage. 

Veil (Crystal). Vacant or flawed region within the 
body of a crystal, caused by unfavorable saturation 
conditions in the solution during the growth of the 
mother bar. 

Window. The portion of a transducer case designed 
to permit the passage of acoustic waves. 

X-CuT (45° X-Cut). a cut in which the electrode 
faces of a piezoelectric crystal are perpendicular to 
its X-, or electric, axis. In the 45° X-cut, the longest 


dimension of the crystal is 45° from the Y and the 
Z axes. 

Y-Cut (45° Y-Cut). a cut in which the electrode 
faces of a piezoelectric crystal are perpendicular to 
its Y-, or mechanical, axis. In the 45° Y-cut, the 
longest dimension of the crystal is 45° from the X 
and the Z axes. 

Z-CUT (45° Z-Cut). a cut in which the electrode faces 
of a piezoelectric crystal are perpendicular to its Z-, 
or optic, axis. In the 45° Z-cut, the longest dimension 
of the crystal is 45° from the X and the Y axes. 




BIBLIOGRAPHY 


Numbers such as Div. 6-611.2-M3 indicate that the document listed has been microfilmed and that its title 
appears in the microfilm index printed in a separate volume. For access to the index volume and to the 
microfilm, consult the Army or Navy agency listed on the reverse of the half-title page. 


Chapter 1 

1. Basic Methods for the Calibration of Sonar Equip- 
ment, Summary Technical Report, Division 6, Vol. 
10 . 

2. Completion Report on Transducer Calibration 
Facilities and Techniques, UCDWR U435. 

3. Principles and Applications of Unde^'water Sotind, 
Summary Technical Report, Division 6, Vol. 7. 

Chapter 2 

1. The Absolute Differential Calculus, Levi-Civita, 
Blackie and Sons, 1927. 

2. Mathematical Theory of Elasticity, A. E. H. Love, 
Cambi’idge University Press, 1934, p. 44. 

2a. Ibid., p. 78. 

2b. Ibid., p. 102. 

2c. Ibid., p. 287 et seq., p. 428. 

3. Lehrbuch der Kristallphysik, W. Voigt, B. G. 
Teubner, 1928. 

3a. Ibid., p. 167. 

3b. Ibid., pp. 218, 965. 

3c. Ibid., pp. 218, 915, 965, 978. 

4. “Impedance Representation of Tangential Bound¬ 
ary Conditions,” G. D. Camp, The Physical Review, 
Vol. 69, May 1946, p. 501. 

5. Vibration and Sound, Philip M. Morse, McGraw- 
Hill Book Co., New York, 1936. 

6. Untersuchungen iiber das Logarithmische und 
Newtonische Potential, C. Neumann, Leipzig, 1877. 

7. “On the Acoustic Shadow of a Sphere,” Lord 
Rayleigh and John William Strutt, baron, Philo- 
sophieal Transactions of the Royal Society 
(London), 1904. 

8. “Uber die von einer starren Kugel hervorgerufene 
Storung des Schallfeldes,” Stenzel, Electrische 
Nachrichten-Technik, Vol. 15, No. 3, 1938. 

9. The Physical Review, G. D. Camp [about Septem¬ 
ber 1946]. 

10. “Absorption of Sound Waves,” Epstein, Theodore 
von Kdrmdn Anniversary Volume, California In¬ 
stitute of Technology. 

11. The Theory of Sotmd, Vol. 2, Lord Rayleigh and 
John William Strutt, baron, Dover Publications, 
New York, 1945. 

12. Electromechanical Transducers and Wave Filters, 
Warren P. Mason, D. Van Nostrand Co., New 
York, 1942, p. 200. 

12a. Ibid., pp. 315, 318. 

12b. Ibid., p. 205. 

13. “Mathematics of the Physical Properties of 
Crystals,” Walter L. Bond, Bell System Technical 
Journal, Vol. 22, January 1943. 

13a. Ibid., p. 22. 


14. “On Dissipative Systems and Related Variational 
Principles,” Bateman, The Physical Review, Vol. 
38, p. 815. 

15. Nature, Bateman, Vol. 131, 1933, p. 472. 

16. Journal fiir die Reine Angewandte Mathematik, 
L. Pochhammer, Vol. 81, 1876, p. 324. 

17. Cambridge Philosophical Society Transactions, C. 
Chree, Vol. 14, 1889, p. 278. 

Chapter 3 

1. Electromechanical Transducers and Wave Filters, 
Warren P. Mason, D. Van Nostrand Co., 1942, p. 
200 et seq. 

la. Ibid., Appendix C. 

lb. Ibid., p. 202. 

2. Dissipation of Energy in Crystal Transducers, 

Parts A to C, Glen D. Camp, File 01.22, UCDWR 

Memorandum, Feb. 19, 1944. Div. 6-611.2-M3 

3. Electrical Measurements for Z-cut ADP 

Crystals—Case 24769-8, H. J. MeSkimmin, Tech¬ 
nical Memorandum MM-44-120-97, BTL, Oct. 6, 
1944. Div. 6-611.1-M7 

3a. Drawing No. BL 382365. 

4. The Design of Crystal Vibrating Systems, William 

J. Fry, John M. Tayloi’, and B. W. Henvis, NRL, 
August 1945. Div. 6-611.2-M6 

5. Constants and Temperature Coefficients of ADP 

Crystals — Case 33222, W. P. Mason and R. G. 
Kinsley, Technical Memorandum MM-43-160-38, 
BTL, Mar. 30, 1943, p. 15. Div. 6-611.1-M5 

5a. Ibid., p. 27. 

6. Properties of 45° Y-cut RS, and the Load Carry¬ 
ing Capacity of Liquids Used with Them, OSRD 
1308, NDRC 6.1-sr346-627, BTL, Dec. 15, 1942, p. 8. 

Div. 6-611.1-M3 

7. Vibration ayid Sound, Philip M. Morse, McGraw- 
Hill Book Co., New York, 1936, p. 123. 

8. Theory of Sound, Lord Rayleigh and John William 
Strutt, baron, Dover Publications, New York, 1945. 

9. Theoretical Physics, Joos, G. E. Stechert and Co. 

10. Acoustics, G. W. Stewart and R. B. Lindsay, D. 
Van Nostrand Co., New York, 1930. 

11. “Der senkrechte und schrage Durchtritt einer in 
einem flussigen Medium erzergten evenen Dila¬ 
tions — Welle durch eine in dieseni Medium 
befindliche planparallele feste Platte,” H. Reissner, 
Helvetica Physica Acta, Vol. 11, 1938, pp. 140-144. 

12. Mechanical and Acoustic Attachments for Piezo¬ 
electric Crystals Used in Transducers, NDRC 
6.1-sr346-628, BTL, Dec. 15, 1942. Div. 6-611.2-M2 

13. Comparison of Two 54 Inch Domes, Eginhard 
Dietze, NDRC 6.1-srll30-1628, Sei'vice Project 
NS-139, CUDWR-USRL, June 19, 1944. 

Div. 6-555-M21 


383 



384 


BIBLIOGRAPHY 


14. Transmission of Sound Through Flat Plates (In¬ 
ternal Memorandum), Edwin M. McMillan, [C4- 
sr30-025] UCDWR, Oct. 15, 1941. Div. 6-611.2-Ml 

15. Technical Data on Plastics, Plastics Materials 
Manufacturei’s’ Association, Washington, D. C. 

Chapter 4 

1. Electromechanical Transducers and Wave Filters, 
Warren P. Mason, D. Van Nostrand Co., New 
York, 1942, p. 205. 

la. Ibid., p. 204 et seq. 

lb. Ibid., p. 207. 

l c. Ibid., p. 235. 

2. Principles and Applications of Underwater Sound, 
Summary Technical Report, Division 6, Vol. 7. 

3. Vibration and Sound, Philip M. Morse, D. Van 
Nostrand Co., New York, 1944, p. 260, Fig. 71. 

4. Methoden der Mathematischen Physik, R. Courant 
and D. Hilbert, Vol. 1, J. Springer, Berlin, 1931, 
p. 412. 

5. Tables and Functions with Formulae and Curves, 
Eugen Jahnke and Fritz Emde, B. G. Teubner, 
Leipzig and Berlin, 3rd Edition, [G. E. Stechert 
& Co., N. Y.] 1938. 

6. Journal of Mathematics, Helmoholtz, Vol. 57, 
1859, p. 7. 

7. Mathematische Annalei}, Vol. 1, 1869, pp. 1-36. 

8. Mathematische Annalen, A. Sommerfeld, Vol. 47, 
1896, p. 317. 

9. Akustische Zeitschrift, K. Osterhammel, Vol. 2, 
March 1941, p. 6. 

10. The Theory of Sound, Vol. 2, Lord Rayleigh and 
John William Strutt, baron, Dover Publications, 
New York, 1945. 

11. “Bei’echnung des Schallfeldes von Kreisforigen 
Kolbenmenbranen,” Electrische Nachrichten- 
Technik, Vol. 4, 1927, p. 239. 

12. Mathematische Annalen, K. Swartschild, Vol. 55, 
1902. 

13. “Das Schallfeld der Kreisforigen Kolbenmen¬ 
branen,” H. Backhaus, Annalen der Physik, Vol. 

5, 1930, p. 1. 

14. “Acoustic and Inertia Pressure at any Point on a 
Vibrating Circular Disc,” N. W. McLachlan, 
Philosophical Magazine, Vol. 14, Ser. 7, 1932, pp. 
1012-1025. 

15. Annalen der Physik, H. Stenzel, Vol. 4, 1942, p. 18. 

16. “Acoustic Radiation Field of Piezoelectric Oscil¬ 
lator,” L. V. King, Canadian Journal of Research, 
Vol. 11, August 1934, p. 135. 

17. Akustische Zeitschrift, K. Menges, Vol. 6, 1941, 
p. 90. 

18. American Journal of Acoustics, R. Clark Jones, 
Ser. 16, 1945, p. 147. 

19. Electrische Nachrichten-Technik, H. Stenzel, Vol. 
4, 1927, p. 239. 

20. Electrische Nachrichten-Technik, H. Stenzel, Vol. 

6, 1929, p. 165. 

21. “The Impedance of Telephone Receivers as Affected 


by the Motion of Their Diaphragm,” Kennelly and 
Pierce, Proceedings of the American Academy of 
Arts and Sciences, September 1912. 

Chapter 5 

1. Radio Engineering, F. E. Terman, McGraw-Hill 
Book Co., New York, 1937, Ch. 7. 

la. Ibid., Sec. 43, Chap. 5; Sec. 55, Chap. 6. 

lb. Ibid., Sec. 61, Chap. 7. 

l c. Ibid., Sec. 5, par. 11; Sec. 3, par. 27. 

2. “Circle Diagram in Tube Circuits,” A. A. Nims, 
Electronics, Vol. 12, No. 3, May 1939, p. 23. 

3. Motion Picture Sound Engineering, D. Van 
Nostrand Co., New York, 1938, Chaps. XVI, XVII. 

Chapter 6 

1. Electromechanical Transducers and Wave Filters, 
Warren P. Mason, D. Van Nostrand Co., New 
York, 1942, p. 202 et seq. 

2. Elements of Acoustical Engineering, Harry F. 
Olsen, D. Van Nostrand Co., New York, 1940. 

Chapter 7 

1. The Design of Crystal Vibrating Systems, William 
J. Fry, John M. Taylor, and B. W. Henvis, U. S. 
Government Printing Office, NRL, August 1945. 

Div. 6-611.2-M6 

Chapter 8 

1. “The Growing of Crystals,” R. W. Moore, J. Am. 
Chem. Soc., Vol. 41, 1919, p. 1060. 

2. U. S. Patent Reissues, 19697 (1935), 19698 (1935), 
B. Kjellgren. 

3. German Patent 228, 246 (1908), F. Krueger and 
W. Finke. 

4. “Dilations in Rochelle Salt,” 1. Vigness, The 
Physical Review, Vol. 48, 1935, pp. 198-202. 

5. Interelectrode Leakage of Rochelle Salt Crystals 

Used in Piezoelectric Devices, H. J. McSkimin, 
Technical Memorandum MM-41-160-52, BTL, Oct. 
14, 1941. Div. 6-611.1-Ml 

6. Data on Piezoelectric PN Crystals (Revised 

Edition), Hans Jaffe, Brush Development Com¬ 
pany, Nov. 23, 1943. Div. 6-611.1-M6 

7. ADP Crystals, I. Growing ADP Crystals and 
Establishment of Pilot Plant, II. Processing ADP 
Crystals, III. Electrical Testing of ADP Crystals, 
NXss-31643 and NXsr-46932, Task Nos. 7 and 8, 
BTL and BuShips, Dec. 15, 1945. Div. 6-611.1-M8 

8. The Electrical Resistivity of Crystals of Ammo¬ 
nium and Potassium Dihydrogen Phosphate, J. B. 
Johnson and H. B. Briggs, Technical Memorandum 
MM-43-160-32, BTL, Mar. 20, 1943. 

Div. 6-611.1-M4 

9. Specification for ADP Crystal Plates, NAVSHIPS 

(940-930D) Radio Division, Navy Department RE- 
13A-922A, Mar. 16, 1944. Div. 6-611.2-M4 




BIBLIOGRAPHY 


385 


10. Piezoelectricity, Walter G, Cady, McGraw-Hill 
Book Co., New York, 1946. 

11. U. S. Patent 1,764,088, C. B. Sawyer; also 
Canadian Patent 302,568. 

12. Electrical Measurement for A5° Z-Cut ADP 
Crystals, H. J. McSkimin, Technical Memorandum 
MM-44-120-97, BTL, Oct. 6, 1944. Div. 6-611.1-M7 

13. Properties of U5° Y-Ciit Rochelle Salt Crystals, 

and the Load Carrying Capacity of Liquids Used 
with Them, OSRD 1308^ NDRC 6.1-sr346-627, 
BTL, Dec. 15, 1942. Div. 6-611.1-M3 

14. Fundamental Studies of X-Cut Rochelle Salt, Darol 

K. Froman, NDRC C4-sr30-392, UCDWR, July 15, 
1942. Div. 6-611.1-M2 

15. Mechanical and Acoustic Attachments for Piezo¬ 

electric Crystals Used in Transducers, NDRC 
6.1-sr346-628, BTL, Dec. 15, 1942. Div. 6-611.2-M2 

16. The Investigation of Adhesives for ADP Crystal 
Assemblies, Bell Telephone Laboratories, Technical 
Memorandum 44-120-99, BTL, Oct. 10, 1944. 

Div. 6-611.2-M5 

17. Letter from C. J. Frosch of Bell Telephone 
Laboratories to S. E. Urban of B. B. Chemical 
Company, Cambridge, Mass., Mar. 3, 1945. 

18. “The Behavior of Nickel Copper Alloys in Sea 
Water,” F. L. LaQue, Journal of the American 
Society of Naval E7igineers, Vol. 53, February 
1941, pp. 29-64. 

19. “Some Observations of the Potentials of Metals 


and Alloys in Sea Water,” F. L. LaQue and G. L. 
Cox, Proceedings of the American Society of 
Testing Materials, Vol. 40, June 1940. 

20. “Brass Plating for Rubber Adhesion,” H. P. Coats, 
Monthly Review, January 1937, pp. 5-26; “Brass 
Plating for Rubber Adhesion,” W. Hayford and 
H. S. Rogers, Mo^ithly Review, May 1945. 

21. Measurmg Tank Suitable for Acoustic Measure¬ 

ments m Water, 6.1-NDRC-836, BTL, Mar. 31, 
1943. Div. 6-553.4-M2 

22. Absor-ption of Coated Steel Plates, Eginhard 
Dietze, NDRC 6.1-srll30-2146, Service Project 
NS-139, CUDWR-USRL, Apr. 5, 1945. 

Div. 6-552-M17 

Chapter 9 

1. Electrical Communication, M. Levy, Vol. 18, 1940, 

p. 206. 

2. Electro7iics, T. P. Tayloi*, Vol. 12, 1934, p. 62. 

3. Fundameyital Studies of X-Cut Rochelle Salt, 

Darol K. Froman, NDRC C4-sr30-392, UCDWR, 
July 15, 1942. Div. 6-611.1-M2 

4. Electromechayiical Transducers aiid Wave Filters, 
Warren P. Mason, D. Van Nostrand Co., New 
York, 1942, p. 203. 

4a. Ibid., p. 205. 

5. Smithsonian Mathematical Formulae, Smithsonian 
Institution, Washington, D. C., 1939, p. 129. 




CONTRACT NUMBERS, CONTRACTORS, AND SUBJECT OF CONTRACTS 


Contract 

Number 

Name and Address 
of Contractor 

Subject of Contract 

OEMsr-30 

The Regents of the University of California 
Berkeley, California 

Maintain and operate certain laboratories 
and conduct studies and experimental in¬ 
vestigations in connection with submarine 
and subsurface warfare. 

OEMsr-346 

Western Electric Company, Inc. 

New York, New York 

Studies and experimental investigations in 
connection with submarine and subsur¬ 
face wai’fare. 


386 







INDEX 


The subject indexes of all STR volumes are combined in a master index printed in a separate volume. 

For access to the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. 


A.\60 transducer cable, 346 
Acheson 1008 (aqueous graphite sus¬ 
pension), 285 
Acoustic ammeter, 7 
Acoustic reciprocity principle, applica¬ 
tion to transducers, 152-161 
calibration of transducers, 133-134 
directivity index, 159-161 
equivalent circuit, 59-61 
lobe suppression, 154-159 
theory, 132 
Acoustic windows 

see Windows, acoustic 
Acoustic-isolation materials, 125-126, 
261-263, 328-330 
Airfoam rubber, 257, 262-263 
applications, 262-263 
Cell-tite foam neoprene, 126 
cellular rubber, 329-330, 346 
cork and cork-rubber compositions, 
330 

Corprene, 125, 262, 265-266 
dichloro-difluoro methane, 328 
Foamglas, 125-126, 262 
free gas, 328-329 
Freon, 328 
metal-air cells, 329 
summary of types, 261-262 
Acryloid B-7 cement 

application technique, 315-316 
attaching tin-foils to crystals, 287 
Admittance, absolute, 162-163 
Admittance curves, 332 
Admittance measurements, transducer 
see Impedance measurements, trans¬ 
ducer 

Admittance measuring circuit 
crystal capacitance, 301-302 
crystal resonance, 304 
Admittance specifications for crystals, 
302-303, 332 
ADP crystals 

see Ammonium dihydrogen phos¬ 
phate crystals 
Airfoam rubber 

applications, 257, 262-263 
comparison with Corprene, 262 
Akron Paint and Varnish Company, 
antifouling paints, 341 
Aluminum faces for transducer ele¬ 
ments, 119, 284 

Amercoat (corrosion-resisting coating), 
340-341 

American Pipe and Construction Com¬ 
pany, Amercoat, 340-341 
Ammeter, acoustic, 7 
Ammonium dihydrogen phosphate 
crystals 

advantages over Rochelle salt, 231-232 


capacitance measurements, 300-301 
characteristic impedance at zero 
width, 89 

dielectric constant, 93 
electromechanical coupling coeffi¬ 
cient, 91-92 
formula, 271-272 

matrix formulation using crystallo¬ 
graphic axes, 55-56 
matrix formulation using rotated 
axes, 56-57 

piezoelectric coupling coefficient, 91- 
92 

solution of boundary-value problem, 
64-72 

summary of constants, 93 
upper temperature limit, 306 
Ammonium dihydrogen phosphate 

crystals, characteristics 
chemical properties, 271-272 
electrical properties, 273-274 
impedance measurements, 89 
surface resistance, 273-274 
thermal behavior, 273 
volume-resistivity, 273 
Ammonium dihydrogen phosphate 

crystals, processing 
bonding to rubber, 310-314 
grinding, 289-291 
growing process, 271-273 
milling, 294-295 
orientation of bars, 277-279 
reflectoriascope, 278 
rough-cutting from bars, 279-280 
sawing, 292-294 
spliced crystals, 281 
surface finishing, 280-281 
Ammonium dihydrogen phosphate 
crystals, specifications 
d-c volume resistivity, 305 
electrical characteristics, 300 
high voltage, 305 
oscillatory characteristics, 304 
Amplifiers, transducer, 73, 217-226 
effect of impedance-matching ratios, 
220-223 

effect of load magnitude and power 
factor, 218-219 

for receiving transducers, 224-225 
guard amplifier, 224-226 
impedance-match, 224 
preamplifier, 74 
pulsed power, 224 
transmitting response, 179 
Amplitude-directivity patterns, 130 
Antifouling paint, 341 
Antiresonance of crystals 
definition, 303 
electrical, 83-84 


mechanical, 82 

Assembly and mounting of trans¬ 
ducers, 95-97, 350-357 
backing plates, 322-323 
Beniofif blocks, 97 
cable installation, 351-352 
cement joints, 95-96 
crystal arrays, 316-319, 322-323, 350- 
351 

crystal blocks, 96-97 
liquid-filling technique, 355 
matching networks and cables, 351 
sealing cases, 352-355 
unit-construction, 97 

B-7 cement, Acryloid 
application technique, 315-316 
attaching tin-foils to crystals, 287 
Backing plates, 75, 98-110 
calculation of flexural modes, 98-99 
calculation of impedance, 103-105, 
360-361 

coupling techniques, 98 
criterion for determining minimum 
thickness, 105-106 

effect on surface-velocity distribu¬ 
tions, 108-110 

frequency of flexure of a rectangular 
bar, 98 

modal patterns for a square plate, 99 
QBF-type, 102 
slotted square bar, 102 
spurious vibrations, 263 
suppression of flexural modes, 101-102 
Backing plates, design, 262-264 
acoustic-isolation materials, 262 
insulation, 264 
mounting technique, 322-323 
multiple layers, 103-106, 263 
thin plates, 264 

use of silicone or Univis oil, 106 
Backing plates, materials, 319-322 
aluminum, 119 
Cerrobend, 110-111, 321 
Duralumin, 322 
glass, 122, 263, 321 
lead, 121, 321-322 
magnesium, 121 
Meehanite, 120 
plastic plates, 321 
steel, 319-321 
Wood’s metal, 121 
Bakelite, use in transducers, 121 
Bakelite cement, use in transducers 
application technique, 307-309 
BC-6052 cement, 123, 286 
effect of transducer filling liquid, 309 
Baker Castor Oil Company, oil filling 
for transducers, 347 


387 


388 


INDEX 


Bandwidth of crystals 
Benioff blocks, 97 

in constant-voltage drive crystals, 
82-83 

BC-6052 Bakelite cement 
coupling of crystals, 123 
electroding crystals, 286 
BE transducer, 4 
Beam patterns 

see Directivity patterns 
Beeswax, use in transducers, 124 
Bell Telephone Laboratories 
Butyl-C cement, 124, 309-310 
electrode of crystals, 93 
jacketed foam rubber, 262 
pc rubber, 264, 336-337 
transducer, 4 
Benioff blocks, 97 
BG transducer, 4 
Blocked crystal 
definition, 78 
resistance, 80 

Blocked impedance of transducers, 161 
Boundary-value problem, solution,64-72 
approximation including all second- 
order moments, 69-72 
Mason approximation, 67-69 
piezoelectrics, 49-51 
steady-state, 61-62 
Brass, use in transducers, 119 
Bridge circuit for crystal capacitance 
measurement, 301 

Bridge methods for transducer imped¬ 
ance measurement, 360-362 
balanced-to-ground arrangement, 
361-362 

effect of cable, 360-361 
“hy-bridge,” 361-362 
metallic backing plate, 360-361 
nonmetallic backing plate, 360 
Schering bridge, 360 
three-terminal impedance, 360 
two-terminal impedance, 360 
Bronze for transducer cases, 119 
Brush Development Company 

milling equipment for processing 
crystals, 294 

synthetic Rochelle salt crystals, 269- 
270 

transducer, 4 

BTL (Bell Telephone Laboratories) 
Butyl-C cement, 124, 309-310 
electrode of crystals, 93 
jacketed foam rubber, 262 
pc rubber, 264, 336-337 
transducer, 4 

Butacite VF-7100 cement, 314 
Butyl phthalate for filling transducers, 
124-125 

Butyl-C cement, use in transducers 
application technique, 309-310 


manufacturing process, 310 
properties, 310 

C-3 Cycle-Weld cement, 123-124 
Cable design, transducer, 73, 214-216, 

346- 347 
AA60; 346 
dielectrics, 214-215 

effect on impedance measurements, 
360-361 

effect on performance, 215-216 
installation, 351-352 
Koroseal, 214-215 
oil-tight, 346-347 
packing glands, 343 
polythene, 214-215 
SA60; 346 

Simplex No. 9061; 346, 361 
specifications and tests, 346 
vinyl chloride plastics, 214-215 
Vinylite, 214-215 

Cadmium, use in transducers, 119 
Calcium chloride, use in transducers, 
125 

Calibration of transducers 
application of data, 357 
complex impedance, 161-164 
directivity patterns, 22-25, 139-152 
impedance measurement methods, 
177-178, 358-362 
reciprocity method, 133-134 
response measurements, 164-168, 
235, 243-246, 251-254 
tests, 22-27 
Cases for transducers 

see Housing for transducers 
Castor oil filling for transducers 
acetylated, 125 
Baker’s DB-grade, 124-125 
characteristics and specifications, 

347- 348 

dehydration, 348-350 
Cavitation limit on transducer, 27, 171- 
172, 213 

Cavity modes in transducers, 113 
CD transducer, 7 

Cell-tite foam neoprene, use in trans¬ 
ducers, 126 
Cell-tite rubber 
acoustic isolation, 329-330 
sound-reflecting pads, 346 
Cellular rubber 
see Cell-tite rubber 

Cellulose acetate butyrate plastic for 
acoustic windows, 117 
Cement joints for crystals, 95-96, 305- 
306 

Cements used in transducers, 123-124, 
305-316 

Acryloid B-7; 287, 315-316 
Bakelite, 123, 286, 307-309 


beeswax and rosin, 124 
Butacite VF-7100; 314 
Butyl-C, 309-310 
Cycle-Weld, 123-124, 312-314 
molten Rochelle salt, 124, 315 
Norace, 315 
sealing wax, 124 
thermoplastic, 281, 314-315 
Ty-ply, 338, 346-347 
urea formaldehyde, 315 
Vinylseal, 281 

Vulcalock, 123, 286, 307-309 
Cements used in transducers, applica¬ 
tion techniques, 305-307 
amount, 306 

condition of cemented surface, 306- 
307 

crystal blocks, 96-97 
curing, 307 
humidity, 307 
inertia-driven crystal, 256 
method, 306 
specifications, 305-306 
Ceramics and glasses used in transduc¬ 
ers, 122-123, 263, 321 
Cerrobend transducer backing plates, 
110-111, 321 

Chicago Pneumatic Tool Company, 
tool for banding rubber cylin¬ 
ders, 353 

Chrysler Corporation, Cycle-Weld ce¬ 
ment, 123-124, 312-314 
Circuits, electronic 

see also Equivalent circuits 
admittance measuring circuit, 301- 
302, 304 

bridge circuit for crystal capacitance 
measurement, 301 
CJ transducer, 7 
Clamped drive crystals, 173-176 
design, 255, 257, 259 
electrical Q, 423 
equivalent circuit, 173-174 
evaluation, 232 
groups of crystals, 257 
impedance, 178 
intensity, 180 

maximum short-circuit current, 181 
mechanical Q, 176 
radiation resistance, 177 
resonant frequency, 175 
Clamped drive transducers 
CPIOZ, 236 
CQ4Z, 236 
CQ8Z, 4, 236, 338 
FG8Z, 236 
GA14Z, 236 
JB4Z, 236-246, 258-259 
Coatings, corrosion-resisting 
Amercoat, 340-341 
antifouling paints, 341 



INDEX 


389 


Copolene for transducer cables, 214-215 
Copper, use in transducers, 119-120 
Corprene 

applications, 125, 265-266 
comparison with Airfoam rubber, 262 
sound-reflecting pads, 346 
use in blanking non-radiating faces of 
crystal, 325-326 
Corrosion-resisting coatings 
Amercoat, 340-341 
antifouling paints, 341 
Coupling between transducer sections 
backing plates, 98 
crystals, 123 
exterior case, 74 

CPIOZ clamped drive transducer, 236 
CQ transducer, cylindrical crystal 
array, 318 

CQ4Z clamped drive transducer, 236 
CQ8Z clamped drive transducer 
acoustic window, 338 
crosstalk level, 4 
dual transducer, 236 
Crosstalk in transducers, 4, 265-266 
Crystal arrays, assembly, 316-319 
cylindrical and curved arrays, 318- 
319 

installation, 350-351 
mounting technique, 322-323 
simple flat arrays, 316-317 
stacked arrays, 319 

Crystal arrays, design, 259-260, 316-333 
acoustic-isolation materials, 328-330 
backing plates, 263-264, 319-323 
cylindrical source, 259 
fronting plates, 323 
lobe suppression, 159, 259, 317-318 
multiple motors, 259-260 
requirements, 234 
rubber diaphragm, 324-325 
stacked arrays, 325-326 
Crystal arrays, inspection techniques, 
330-333 

admittance, 332 
capacitance, 332 
d-c resistance, 332 
high voltage, 333 
polarity of crystals, 331-332 
visual inspection, 330-331 
Crystal arrays, wiring techniques, 326- 
328 

choice of materials, 326 
electric contact strips, 326-327 
soldering precautions, 327 
wiring arrangements, 327-328 
Crystal blocks, design, 257-259 
cementing techniques, 96-97 
clamped drive, 255, 257, 258 
directivity patterns, 258 
foiling, 258 
inertia drive, 257-258 


size, 75 

symmetric drive, 258 
Crystal transducers 
see also Crystals 
applications, 27-29 
calibration tests, 22-27 
piezoelectricity, 1-4, 47-59 
power limitations, 27, 213 
typical units, 4-21 

Crystal transducers, component parts, 
73-126 

acoustic windows, 113-119, 123, 336- 
338 

amplifiers, 74, 217-226 
backing plates, 75, 98-106, 263, 360- 
361 

cables, 214-216, 343, 346-347, 360-361 
case, 74-75, 333-350, 352-355 
crystal blocks, 75, 96-97, 257-259 
electronic system, 73 
inert materials, 119-126, 261-263, 
319-321, 328-330 

liquids for filling, 124-125, 309, 347- 
350, 355 

matching network, 73-74, 351 
motor, 75, 107-112, 259-260 
single crystal plates, 75-95 
subassemblies, 95-97 
Crystal transducers, construction tech¬ 
niques, 267-357 

assembly and mounting, 95-97, 322- 
323, 350-357 
cements, 305-316 
GD28; 108-110 

housings and accessories, 333-350 
manufacturing requirements, 231 
precautions when handling crystals, 
267-268 

preparation of arrays, 316-333 
preparation of individual crystals, 
269-297 

storage conditions for crystals, 274- 
275 

Crystal transducers, design, 211-266 
see also Materials used in transducers 
application of Green’s functions, 38- 
41 

application of Neumann boundary- 
value problem, 38-39 
application of reciprocity principle, 
59-61, 132-134, 152-161 
backing plates, 103-106, 262-264, 
322-323 

choice of basic design, 232-233 
choice of crystal material, 231-232 
clamped drive, 236-246, 258, 338 
crosstalk, 4, 265-266 
crystal array, 234, 259-260, 316-333 
crystal blocks, 75, 96, 257-259 
exterior case, 234-235 
groups of crystals, 257-259 


inertia drive, 246-249 
isolation material, 261-263 
lobe suppression, 142-145, 154-159, 
259, 317-318 

response requirements, 235 
single crystals, 233-234, 255-257 
symmetric drive, 249-254 
tangential motion, 260-261 
windows, 264-265 

Crystal transducers, electronic design, 
73, 211-229 
amplifiers, 74 

cables, 214-226, 343, 346-347, 360-361 
characteristics, 211-214 
equalizing networks, 226-229 
matching networks, 73-74, 216-217, 
351 

Crystal transducers, general types, 4-7 
see also UCDWR transducers 
Bell Telephone Laboratories, 4 
Brush Development Company, 4 
clamped drive, 173-174, 236-246, 258, 
338 

cylindrical, 134-139 
inertia drive, 173-174, 236, 246-249 
Submarine Signal unit, 4-7 
symmetric drive, 173-174, 236, 249- 
254 

unit-construction transducers, 97 
Crystal transducers, inspection tech¬ 
niques, 355-357 

see also Testing apparatus for trans¬ 
ducers 

calibration, 357 
d-c resistance, 355-356 
leaks, 355-356 

Crystal transducers, performance limi¬ 
tations, 168-172 
cavitation, 27, 171-172, 213 
power, 27, 213 
receiving, 171 
transmitting, 168-171 
Crystal transducers, plane-radiating, 
139-145 

directivity calculations, 140-142 
lobe suppression, 142-145 
sound field measurements, 142 
Crystal transducers, properties, 127- 
210 

analysis of equivalent circuits, 127- 
128, 173-182 
directivity, 139-152 
electrical characteristics, 211-214 
impedance, 85-87, 161-164, 177-178, 
358-362 

modes, 98-102, 110-113, 257 
possible measurements, 128-129 
radiation theory, 129-139 
reciprocity, 152-161 
resonance, 112-113, 211-212 
response, 138-139, 164-168, 171, 235 





390 


INDEX 


Crystal transducers, specific models 
BE, 4 
BG, 4 
CD, 7 
CJ, 7 

CPIOZ, 236 

CQ4Z, 236 

CQ8Z, 4, 236, 338 

CY4; 7, 236, 249-254, 256 

EP, 7, 323-324 

FE2Z, 236, 246-249 

FG8Z, 236 

GA14Z, 236 

GD, 4 

GD22; 152 

GD28; 108-110 

GD34Z, 236 

JB, 4 

JB4Z, 236-246, 258-259 
JC2Z1; 110-112 
JK, 354 
KC, 4 
KC2; 236 
XCY8-1; 260-261 
Crystal transducers, specifications 
CY4; 249-250 
directivity, 230-231 
general requirements, 230-231 
impedance, 230 
JB4Z, 241 

manufacturing requirements, 231 
power, 230 
response, 230 
Crystals, 75-95 

see also Ammonium dihydrogen phos¬ 
phate crystals; Rochelle salt 
crystals 

electrodes, 93-95, 120, 282-287 
equivalent circuits, 77-87, 93, 173-174 
handling precautions, 267-268 
mounting techniques, 97 
processing techniques, 288-297, 305- 
306 

storage conditions, 274-275 
weakened crystals, 95 
Crystals, capacitance measurements, 
300-304 

admittance measuring circuits, 301- 
302 

ammonium dihydrogen phosphate 
crystals, 300 
bridge circuit, 301 
dependent factors, 300 
formula for capacity ratio, 303-304 
frequency used, 300 
Rochelle salt crystals, 300-301 
Crystals, design 

see also Ammonium dihydrogen 
phosphate crystals, processing; 
Rochelle salt crystals, process¬ 
ing 


clamped drive, 173-176, 178, 255, 
257, 258 

coupling techniques, 123 
inertia drive, 173-178, 232-233, 256, 
262-263 

optical orientation method for crystal 
bars, 278-279 

polarization, 287-288, 296, 331-332 
size of crystal, 233-234 
symmetric drive, 80, 173-180, 256, 258 
weakened crystals, 256-257 
Crystals, evaluation of constants, 87-93 
dielectric constant, 92-93 
effect of electrodes, 93-95 
piezoelectric coupling coefficient, 91- 
92 

resonance frequencies, 87-91 
summary, 93 

Crystals, properties, 268-275 
bandwidth, 82, 97 
chemical properties, 268-271 
crystallizing bars, 269, 272 
electrical properties, 270-271, 273 
impedance, 95 
low-frequency limit, 84-85 
resonance, 81-82, 108 
resonant frequency, 87, 175, 256-257, 
303-305 

storage conditions, 274-275 
tangential motion, 260-261 
temperature limits, 306 
thermal behavior, 270, 273 
Crystals, specifications, 297-305 
admittance and Q, 302-303, 332 
capacitance, 300-302 
cement joints, 305-306 
d-c resistance, 305 
electrodes, 282 
geometric tolerances, 298 
high voltage, 305 
orientation, 298-300 
resonant frequencies, 303-305 
visible defects, 297 
Current-receiving response, 167, 181 
Current-transmitting response, 165- 
166, 179 

CY4 symmetric drive transducer, 7, 
236, 249-254 
basic design, 250 
crystal, 250 

directivity pattern, 250-251, 256 
mechanical details, 251 
response, 251-254 
specifications, 249-250 
Cycle-Weld cement, use in transducers 
application technique, 312-314 
C-3; 123-124 

Cylindrical source, radiation resistance, 
259 

Cylindrical transducers, radiation the¬ 
ory, 134-139 



directivity patterns, 135-136 
impedance, 136, 138 
incomplete arc, 136-138 
response, 138-139 

Damped resonance in transducers, 
112 

Desiccants used in transducers, 125 
Design factors for transducers 
see Crystal transducers, design 
Diaphragms for crystal arrays, 324- 
325 

Dichloro-difluoro methane for acoustic 
isolation, 328 
Dielectrics, theory, 44-47 
constant for crystals, 92-93 
dipole distribution, 44-46 
dissipation, 50-51 
linear, 46-47 

Dielectrics for transducer cables, 214- 
215 

Dimethyl phthalate for filling trans¬ 
ducers, 347-348 

Dipole distribution in dielectrics, 44-46 
Directivity index 

calculation by reciprocity method, 
159-161 

mathematical expression, 131 
Directivity of transducers, 139-152 
amplitude directivity, 130 
directivity factor, 22-25, 131, 148 
effect of case, 152 

effect of surface conditions, 151-152 
phasing, 145-148 
plane radiators, 139-145 
reciprocity theorem, 133 
specifications, 230-231 
theory, 40-41, 139 
Directivity patterns 
crystal blocks, 258 
CY4 transducer, 250-251, 256 
cylindrical transducer, 135-136 
effect of pc rubber, 116 
GD22 transducer, 152 
intensity directivity, 22, 130 
pressure directivity, 22 
Direct-reading phase meter, 364-368 
accuracy, 367 
calibration, 367-368 
flip-flop circuit, 364-367 
Dow Corning fluids for filling trans¬ 
ducers, 125, 348 
Duralumin backing plates, 322 

Echo ranging, transducer requirements, 
28-29 

Efficiency of transducers, 159-161 
Elastics, nonviscous fluids, 37-42 
energy density and flux, 41-42 
field equations; boundary conditions, 
37 



INDEX 


391 


Green’s functions, 38-41 
Neumann boundary-value problem, 
38-39 

steady-state, 37-38 

Elastics, theory, 30-44 

application of Hooke’s law, 35-36 
boundary conditions, 36-37 
displacement and strain, 31-34 
elastic stresses (equations of motion), 
34-35 

energy density (generalized Hooke’s 
law), 35-36 

intensity of crystal radiating into 
water, 30-31 
isotropic solids, 37 
physical principles, 31 
propagation of waves in a crystal¬ 
line medium having no sources, 
35 

Elastics, viscous fluids, 42-44 
reflection conversion, 43-44 
steady-state boundary-value prob¬ 
lem, 42-43 

tangential impedance, 43 

Electric analogues for crystals and 
transducers 
see Equivalent circuits 

Electric network simulator (testing 
apparatus), 373-379 
inertia-driven, 374 
steel backing plate, 379 
transcendental impedance approxi¬ 
mations, 373-374 

Electrical antiresonance of crystals, 
83-84 

Electrical Q of transducer, 175-176, 
212-213 

Electrodes for crystals, 282-287 
aluminum, 284-285 
effect on crystal constants, 93-95 
evaporated, 120, 282-284 
foils, 282, 285-286 
gold, 120, 282-284 
graphite, 285 
silver, 121, 284-286 
specifications, 282 
sprayed, 284-285 
tin-foil, 286-287 
use of cements, 286 

Electromechanical coupling coefficient, 
91-92 

Electronic system of transducer, 211- 
229 

amplifiers, 74, 217-226 
cables, 214-216, 343, 346-347, 360- 
361 

characteristics, 211-214 
equalizing networks, 226-229 
matching networks, 73-74, 216-217, 
351 

Energy density in piezoelectrics, 48-49 


EP transducer 

rubber diaphragm, 323-324 
window-coupled unit, 7 
Equivalent circuits for crystals, 77-87, 
93, 173-174 

clamped drive, 173-174 
constant-voltage bandwidth, 82-83 
electrical antiresonance, transforma¬ 
tion ratio, 83-84 
inertia drive, 173-174 
loaded rectangular crystal, 77-78 
low-frequency limit, 84-85 
mechanical arm, 78-79 
mechanical resonance and antireso¬ 
nance, 81-82 

resistance and reactance, 79-81 
series-equivalent impedance, 85-87 
symmetrically driven, 173-174 
Equivalent circuits for transducers, 
173-182 

see also Network equivalent, trans¬ 
ducer; Network simulator, trans¬ 
ducer 

absolute magnitudeof impedance, 181 
electrical Q, 175-177 
frequency, 175 
impedance, 177-178 
intensity, 180 
Mason circuit, 173 
mechanical Q, 176-177 
nu merical values ofconstants,181-182 
peak open-circuit voltage, 180 
peak short-circuit current, 181 
power factor, 181 
receiver responses, 180 
reciprocity principle, 59-61 
resistance, 177 
short-circuit response, 181 
theory, 127-128 
three basic drives, 173-174 
transmitter responses, 178-179 
Equivalent variational principle, 61-72 
boundary-value problem, 64-72 
steady-state boundary-value prob¬ 
lem, 61-62 
theory, 62-64 

F9-5 pc rubber, 264, 337 
Facings, transducer 

see Materials used in transducers 
FE2Z inertia drive transducer, 236, 
246-249 

crystals, 247-248 
design details, 248 
response, 248 
FG8Z transducer, 236 
Fillings for transducers 

see Liquids for filling transducers 
Flexural modes in transducers, 98-102 
calculation, 98-99 
effect on overall response, 110-112 



groups of crystals, 257 
suppression, 101-102 
Fluids for filling transducers 

see Liquids for filling transducers 
Foam rubber, 261-263 
Airfoam, 257, 262-263 
Cell-tite, 329-330, 346 
Cell-tite foam neoprene, 126 
elimination of cavity modes in trans¬ 
ducer, 113 

Foamglas, use in transducers, 125-126, 
262 
Formulas 

ammonium dihydrogen phosphate 
crystals, 271-272 

capacity ratio of crystals, 303-304 
directivity factor, 22-25, 148 
Rochelle salt, 268-269 
Four-terminal impedance, 163-164 
45° Y-cut RS 

see Rochelle salt crystals 
45° Z-cut ADP 

see Ammonium dihydrogen phos¬ 
phate crystals 
Free-free crystal 
definition, 78 
resonance frequencies, 87 
Freon for acoustic isolation, 328 
Frequency, resonant 

see Resonant frequency of crystals, 
measurement 

Frequency limit, low, of crystals, 84-85 
Frequency response of transducers 
see Response of transducers 

GA14Z transducer, 236 
Galvanic series for sea water, 334 
GD transducer, 4 

GD22 transducer, directivity pattern, 
152 

GD28 transducer 

construction, 108-110 
effect of backing plate on surface- 
velocity distributions, 108-110 
relative velocity of each crystal at 
resonance, 108 
GD34Z transducer, 236 
Geon for transducer cables, 214-215 
German silver, use in transducers, 121 
Glass backing plates 
advantages, 122, 263 
construction, 321 

Gold electrodes for crystals, evaporated, 
120, 282-284 

Goodrich Company, B. F. 
compound 8388 rubber, 337 
pc rubber, 123, 336 
Vulcalock, 123 

Graphite electrodes for crystals, 285 
Green’s functions, application to trans¬ 
ducer design, 38-41 




392 


INDEX 


Hooke’s law, application to elastics, 35- 
36 

Housing for transducers, 333-350 
acoustic-isolation material, 262 
bronze cases, 119 

corrosion-resisting coatings, 340-341 
coupling fluids, 74 
design, 234-235 
effect on directivity, 152 
function, 74 
metal castings, 335 
modes, 112-113 
rubber cases, 338-340 
seals, 341-344, 352-355 
sound-absorbing and -reflecting pads, 
344-346 

tin-can cases, 335-336 
Housing for transducers, specifications, 
333-335 

corrosion resistance, 334-335 
leakage, 333-334 
materials, 333-334 
mechanical strength, 334 
Huygens-Fresnel principle, application 
to transducer design, 39 
Hy-bridge (impedance bridge), 361-362 
Hydraulic gaskets for sealing trans¬ 
ducer cases, 341, 354-355 
Hydrophone, equivalent circuit for 
low-frequency limit, 85 

Impedance measurements, transducer, 
358-362 

absolute magnitude, 162-163, 181, 
358-359 

boundary conditions, 36 
bridge methods for two- and three- 
terminal networks, 360-362 
circuit requirements, 359 
comparison with capacitor, 212-213 
effect of cable, 360-361 
equivalent circuits, 177-178 
rectangular crystal, 77 
series-equivalent impedance, 27, 85- 
87 

specifications, 230 
Young’s modulus, 87-91 
Impedance of symmetrically-driven 
crystals, 178 

Impedance quantities characteristic of 
transducers 

blocked impedance, 161 
motional impedance, 161-162 
radiation impedance, 40-41, 131-132, 
177, 259 

tangential impedance, 43 
Impedance transforming network, 217 
Inertia-driven crystals, design 
cementing requirements, 256 
determination of size, 233 
equivalent circuit, 173-174 


gas-filled unit, 262 
liquid-filled unit, 262 
Inertia-driven crystals, properties 
electrical Q, 176 
evaluation, 232-233 
groups of crystals, 257-258 
impedance, 178 
intensity, 180 

maximum short-circuit current, 181 
mechanical Q, 176-177 
resistance, 80-81, 177 
resonant frequency, 175 
Inertia-driven transducers 
FE2Z, 236, 246-249 
GD34Z, 236 

rubber diaphragm, 323-325 
simulated network, 374 
Intensity directivity pattern, 22, 130 
Intensity radiated from crystal faces, 
30-31, 180 

Inverse piezoelectric effect, 1 
Iron, use in transducers, 120-121 
Isolation materials for transducers 
see Acoustic-isolation materials 
Isotropic solids, theory, 37 

JB transducer, 4 
JB4Z transducer, 236-246 
choice of crystals, 241 
crystal size and shape, 241-242 
crystal spacing, 258 
design details, 241-242 
response, 243-246 
specifications, 241 

JC2Z1 transducer, response, 110-112 
JK transducer, 354 

KC transducer, 4 

KC2 transducer, 236 

Koroseal for transducer cables, 214-215 

Lead backing plates, 121, 321-322 
Liquids for filling transducers, 347-350 
butyl phthalate, 124-125 
castor oil, 124-125, 347-350 
characteristics and specifications, 
347-348 

dehydration, 348-350 
dimethyl phthalate, 347-348 
effect on cements, 309 
filling technique, 355 
olive oil, 347-348 
petroleum oils, 125 
silicones, 106, 125, 348 
Ucon oil, 348 

Listening, transducer requirements, 27- 
28 

Lobe suppression in transducers, 154-159 
by crystal spacing, 145 
design requirements for crystal array, 
259, 317-318 



effect on radiated power, 154-156 
effect on receiving response, 158 
method for circular arrays, 159 
plane radiators, 142-145 
series-parallel arrangements, 158 
Lucite acoustic windows, 116-117, 122 

M-163 pc rubber, 336-337 
Magnesium backing plates, 121 
Manufacturing requirements for trans¬ 
ducers, 231 
Mason circuit 

approximation for boundary-value 
problem, 67-69 

equivalent for transducers, 173 
Matching netw'orks for transducers 
design, 73-74, 216-217 
installation, 351 
Materials, crystal 

see Ammonium dihydrogen phos¬ 
phate crystals; Rochelle salt 
crystals 

Materials used in transducers 

acoustic-isolation materials, 125-126, 
261-263, 265-266, 328-330 
cements, 95-96, 123-124, 287, 305-316 
desiccants, 125 

galvanic series for sea water, 334 
glasses and ceramics, 122-123, 263,321 
liquids, 124-125, 309, 347-350 
magnesium, 121 
metals, 117-121, 319-321 
plastics, 117, 121-122, 321 
rubber, 114-116, 324-325, 329-330, 
336-340 

Mechanical antiresonance of crystals, 82 
Mechanical Q of transducers, 176-177 
Mechanical resonance of crystals, 81-82 
Meehanite backing plates, 120 
Metal castings for transducer housings, 
335 

Metals used in transducers, 119-121 
aluminum, 119, 284 
brass, 119 
bronze, 119 
cadmium, 119 
Cerrobend, 110-111, 321 
copper, 119-120 
Duralumin, 322 
German silver, 121 
gold, 120, 282-284 
iron, 120-121 
lead, 121, 321-322 
Meehanite, 120 
silver, 121, 284-286 
steel, 117-119, 319-321 
tin, 121, 286-287, 335-336 
Wood’s metal, 121 
zinc, 121 

Modes in transducers, 110-113 
cavity, 113 




INDEX 


393 


flexural, 98-102, 110-112, 257 
parasitic, 112-113 
vibrational, 98 
Molecular piezoelectricity, 1 
Monel screen (sound-absorbing pad), 
345 

Motional impedance, transducer, 161- 
162 

Motor of transducer, probe examina¬ 
tion techniques, 75, 107-112 
GD28 transducer, 108-110 
JC2Z1 transducer, 110-112 
measurements in air and oil, 107 
multiple, 259-260 
velocity distribution, 107 
Mountings for transducers 

see Assembly and mounting of trans¬ 
ducers 

Naval Research Laboratory 
mounting of crystals, 97 
pc rubber, 264, 337 
Neoprene 

acoustic windows, 116, 265 
Cell-tite foam neoprene, 126 
Network equivalent, transducer 

see also Equivalent circuits for trans¬ 
ducers 

equalizing, 226-229 
impedance transforming network, 217 
matching, 73-74, 216-217, 351 
power factor correcting network, 216 
two- and three-terminal, 360-362 
Network simulator, transducer, 373- 
379 

inertia-driven, 374 
steel backing plate, 379 
transcendental impedance approxi¬ 
mations, 373-374 

Neumann boundary-value problem, 
application to transducer de¬ 
sign, 38-39 

Norace cement, application technique, 
315 

Norton Company, Norace cement, 315 
NRL (Naval Research Laboratory) 
mounting of crystals, 97 
pc rubber, 264, 337 
Nylon acoustic windows, 117 

Oil filling of transducers 
castor oil, 124-125, 347-350 
olive oil, 347-348 
petroleum, 125 
Ucon oil, 348 

Oil plugs for sealing transducer cases, 
343-344 

Olive oil, disadvantages for filling trans¬ 
ducers, 347-348 

Open-circuit response, transducer, 166- 
167, 180 


Optical orientation method for crystal 
bars, 278-279 

0-ring hydraulic gaskets for sealing 
transducer cases, 341, 354-355 

Paints, antifouling, 341 
Parasitic modes in transducers 
case modes, 112-113 
cavity modes, 113 
types of resonance, 112-113 
Petroleum oils for filling transducers, 
125 

Phase meter, direct-reading, 364-368 
accuracy, 367 
calibration, 367-368 
flip-flop circuit, 364-367 
Phase variation of transducers, 145-148 
Phenomenological piezoelectricity, 1 
Piezoelectric coupling coefficient of 
crystals 

ammonium dihydrogen phosphate, 
91-92 

Rochelle salt, 91-92 

Piezoelectrics, matrix formulations, 51- 
59 

ammonium dihydrogen phosphate 
crystals, 55-57 

matrices for rotated cuts, 56-59 
Rochelle salt crystals, 54-59 
strain matrix, 52 
stress matrix, 52 

symmetry reduction of the matrices, 
54-56 

Piezoelectrics, theory, 1-2, 47-59 
direct and inverse effect, 1 
energy density, 48-49 
equations of propagation, 49 
equations of state, 48-49 
internal viscous and dielectric dis¬ 
sipation, 50-51 
molecular, 1 
phenomenological, 1 
surface dissipation, 49-50 
Pittsburgh-Coming Glass Company, 
foamglas, 125-126 
Plastics used in transducers 
Bakelite, 121 
Lucite, 116-117, 122 
Plexiglas, 122 
polystyrene, 122 

Polythene (polyethylene), 122, 214- 
215 

Tenite If; 117 
Plexiglas, properties, 122 
PN crystals 

see Ammonium dihydrogen phos¬ 
phate crystals 
Polarization, crystal 
equipment, 296 
inspection, 331-332 
marking, 288 


technique, 287-288 

Polystyrene, advantages for trans¬ 
ducers, 122 

Polythene (polyethylene) 

for transducer cables, 214-215 
properties, 122 

Porcelain enamel, insulating material 
for transducers, 122 
Power factor correcting network, 216 
Power limitations, transducer, 27, 213 
Power output of transducers 

effect of lobe suppression, 154-156 
equivalent circuit, 181 
specifications, 230 
transmitting response, 166 
Power requirements for transducers, 
230 

Pressure directivity pattern, 22 
Probe microphone for testing trans¬ 
ducers, 7, 362-364 

see also Motor of transducer, probe 
examination techniques 
Pulse modulator, 368-371 
Pulsed power amplifier, 224 
Punch-Lok Company, transducer 
bands, 353 

Q of crystal 

electrical, 175-176 
mechanical, 176 
specifications, 302-303 
Q of transducer, 175-177 
electrical, 175-176, 212-213 
mechanical, 176-177 
QBF-type-backing plates, 102 

Radiation impedance, 131-132 
clamped drive crystals, 177 
cylindrical source, 135-136, 138, 259 
dependent factors, 132 
method of measuring, 132 
symmetrically-driven crystals, 177 
theory of calculations, 40-41 
Radiation patterns 

see Directivity patterns 
Radiation theory, 129-139 

cylindrical transducers, 134-139 
reciprocity, 132-134 
sound field, 129-131 
Reactance of crystals, equivalent cir¬ 
cuit, 79-81 
blocked, 80 
inertia drive, 80-81 
Receiving patterns 

see Directivity patterns 
Receiving response of transducers 
effect of lobe suppression, 158 
equivalent circuits, 180 
limitations, 171 
matched-receiver, 167 
open-circuit, 166-167 





394 


INDEX 


short-circuit, 167 
voltage response, 166-167, 180 
Reciprocity principle, application to 
transducers, 132-134, 152-161 
calibration of transducers, 133-134 
directivity, 133 
directivity index, 159-161 
equivalent circuit, 59-61 
lobe suppression, 154-159 
theory, 132 

Rectangular crystal, equivalent circuit, 
77-78 

Reflectoriascope for optical orientation 
of crystal bars, 278 

Resistance of crystals, equivalent cir¬ 
cuit, 79-81 
blocked, 80 
inertia drive, 80-81 
symmetric drive, 80 
Resistance of transducer, equivalent 
circuit, 177 
Resonance of crystals 
mechanical, 81-82 
relative velocity at resonance, 108 
Rochelle salt, 98 

Resonance of transducers, 112-113, 211- 
212 

Resonant frequency of crystals, meas- 
ment 

admittance measuring circuit, 304 
clamped drive, 175 
definition, 303 
free-free crystal, 87 
inertia drive, 175 
method of lowering, 256-257 
specifications, 303-305 
symmetric drive, 80 
symmetrically driven crystals, 175 
Response of transducers 
calibration tests, 22-25 
curves, 212 

design requirements, 235 
effect of flexural modes, 110-112 
receiving, 158, 166-167, 171, 180, 181 
short-circuit, 167, 181 
specifications, 230 
system response, 167-168 
transient response, 168 
transmitting, 164-166, 178-179 
pc rubber, 336-337 

acoustic windows, 114-116, 123, 264 
Bell Telephone Laboratories (M-163 
rubber), 336-337 

B. F. Goodrich Company (79-SR-32 
rubber), 121, 336 

effect on directivity patterns, 116 
Naval Research Laboratory (F9-5 
rubber), 264, 337 

Rochelle salt, molten (cement), 124, 315 
Rochelle salt crystals 
capacitance measurements, 300-301 


comparison with ammonium di¬ 
hydrogen phosphate, 231-232 
formula, 268-269 

matrix formulations using crystallo¬ 
graphic axes, 54-56 
matrix formulations using rotated 
axes, 57-59 

solution of boundary-value problem, 
64-72 

summary of constants, 93 
synthetic, 269-270 

X-cut, use as research tool, 371-373 
Rochelle salt crystals, characteristics 
chemical properties, 268-269 
dielectric constant, 92-93 
dimensions required for resonance at 
40 kc, 98 

electric properties, 270 
electromechanical coupling coeffi¬ 
cient, 91-92 
impedance, 212-213 
leakage conduction, 273-274 
thermal behavior, 270 
upper temperature limit, 306 
Rochelle salt crystals, processing 

factors controlling successful crystal¬ 
lization, 269-270 
grinding, 288-289 
milling, 294 

orientation of bars, 275-276 
rough-cutting from bars, 276 
sawing, 291-292 
spliced crystals, 281 
surface finishing, 277 
Rohm and Haas, Acryloid B-7 cement, 
287, 315-316 

Rosin, use in transducers, 124 
RS crystals 

see Rochelle salt crystals 
Rubber acoustic windows, 336-338 
acceptance test, 337 
advantages, 114-116 
compound 8388; 337 
deterioration, 340-341 
metal bonds, 337-338 
neoprene, 116, 265 
pc rubber, 114-116, 123, 336-337 
steel reinforcements, 116 
Rubber cases, transducer, 338-340 
cylindrical cases, 338-339 
molded-rubber cases, 339 
steel-reinforced rubber, 339-340 
Rubber diaphragms for crystal arrays, 
324-325 

Rubber materials used in transducers 
Airfoam rubber, 257, 262-263 
Cell-tite, 126, 329-330, 346 
compound 8388; 337 
pc rubber, 114-116, 123, 264, 336- 
337 

SA60 transducer cable, 346 



Schering impedance bridge, 360 
Sealing methods for transducer cases, 

341- 344, 352-355 

banding rubber cylinders, 352-353 
crimp-sealing, 355 
gaskets, 341, 353-354 
glass-metal terminal seals, 342-343 
oil plugs, 343-344 

0-ring hydraulic gaskets, 341, 354- 
355 

Punch-Lok bands, 353 
Sensitivity of transducers 

see Receiving response of trans¬ 
ducers 

Short-circuit response, transducer 
equivalent circuit, 181 
receiving, 167 

Silica gel, use in transducers, 125 
Silicones for filling transducers, 106, 
125, 348 

Silver foil electrodes for crystals, 121, 
284-286 

Simplex No. 9061 transducer cable, 346, 
361 

Slotted square bar backing plates, 
102 

Sound field, radiation theory, 129-131 
pressure, 130 
radiation zone, 130 
Sound field measurements, 142 
Sound-absorbing pads, 344-346 
Sound-reflecting pads, 346 
Sound-water rubber, 336-337 

acoustic windows, 114-116, 123,'264 
Bell Telephone Laboratories (M-163 
rubber), 336-337 

B. F. Goodrich Company (79-SR-32 
rubber), 123, 336 

effect on directivity patterns, 116 
Naval Research Laboratory (F9-5 
rubber), 264, 337 
Specifications 

cables for transducers, 346 
crystals, 282, 297-306 
filling liquids for transducers, 347- 
348 

housing for transducer, 333-335 
transducers, 230-231, 241, 249-250 
Sperti Incorporated, glass-metal ter¬ 
minal seals for transducer cases, 

342- 343 

Sponge Rubber Products Company, 
Cell-tite foam neoprene, 126 
Stack transducers, crystal arrays 
assembly, 319 
design, 325-326 

Steel acoustic windows, 117-119 

angular variation in transmission, 
118 

double-layer construction, 118-119 
transmission loss, 118 



INDEX 


DECLASSTFTED 
3y authority Secretary <395 


Steel backing plates, construction, 319- 
321 

Steel-reinforced rubber transducer 
cases, 339-340 
Storage of crystals, 274-275 
Strain matrix for piezoelectrics, 52 
Stress formulas, elastic, 34-35 
Stress matrix for piezoelectrics, 52 
Stupakoff Ceramic and Manufacturing 
Company, glass-metal terminal 
seals for transducer cases, 342- 
343 

Submarine Signal transducer, 4-7 
Symmetrically-driven crystals, 173-180 
design, 256 
electrical Q, 175-176 
equivalent circuit, 173-174 
groups of crystals, 258 
impedance, 178 
intensity, 180 
mechanical Q, 176 
radiation resistance, 177 
resistance, 80 
resonant frequency, 175 
Symmetrically-driven transducers 
CY4: 7, 236, 249-254, 256 
KC2; 236 

Synthetic Rochelle salt crystals, 269- 
270 

Synthetic rubber (neoprene) 
acoustic windows, 116, 265 
Cell-tite foam neoprene, 126 

Tangential impedance of crystals, 43 
Tangential motion in crystals, 260- 
261 

Temperature limits of crystals, 306 
Tenite acoustic windows, 117 
Terminal box seals, transducer 

see Sealing methods for transducer 
cases 

Testing apparatus for transducers, 316- 
379 

see also Crystal transducers, inspec¬ 
tion techniques 

direct-reading phase meter, 364-368 
electric network simulator, 373-379 
probe microphone, 7, 362-364 
pulse modulator, 368-371 
use of 45-degree X-cut RS as a re¬ 
search tool, 371-373 
Theory of crystal transducers 
see Piezoelectrics, theory 
Thermoplastic cements, 314-315 
Butacite VF-7100; 314 
Vinylseal, 281 

Three-terminal networks, transducer, 
163-164, 360-362 


Tin, use in transducers 
crystal electrodes, 286-287 
metal plating, 121 
transducer housing, 335-336 
Transducers 

see Crystal transducers 
Transmitting limitations of trans¬ 
ducers, 168-171 

partially loaded transmitters, 170-171 
short pings, 170 
steady-state operation, 168-170 
Transmitting patterns 
see Directivity patterns 
Transmitting response of transducers, 
164-166 

constant-current, 165-166, 179 
constant-power, 166 
constant-voltage, 164-165, 179 
equivalent circuits, 178-179 
idealized amplifier, 179 
Two-terminal netw'orks, transducer, 
163, 360-362 

Ty-ply, use in transducers, 338, 346-347 
UCDWR 

acoustic ammeter, 7 
probe microphone, 7, 362-364 
UCDWR transducers 
BE, 4 
BG, 4 

CD and CJ, 7 

CPIOZ, 236 

CQ, 318 

CQ4Z, 236 

CQ8Z, 4, 236, 338 

CY4; 7, 236, 249-254, 256 

EP, 7, 323-324 

FE2Z, 236, 246-249 

FG8Z, 236 

GA14Z, 236 

GD, 4 

GD34Z, 236 

JB, 4 

JB4Z, 236-246, 258 
KC, 4 
KC2: 236 

Ucon oil 50-HB-100 for filling trans¬ 
ducers, 348 

Underwater transducers 
operation, 1 
purpose, 1 

types of crystals, 1-2 
Union Carbide and Carbon Company, 
Ucon oil, 348 

Unit-construction transducers, 97 
University of California Division of 
War Research 
UCDWR 


Univis oil^^sP ii^trariMfi^ backing 
plates, 106 

Urea formaldehyde cement, application 

August 1960 

transducers, 61-72 
boundary-value problem, 64-72 
steady-state boundary-value prob¬ 
lem, 61-62 
theory, 62-64 

Vibrational modes in transducer, 98 
Vinyl chloride plastics for transducer 
cables, 214-215 

Vinylite for transducer cables, 214-215 
Vinylseal cement for consolidating 
transducers, 281 

Voltage receiving response, 166-167, 
180 

Voltage transmitting response, 164-165, 
179 

Vulcalock cement, use in transducers 
application technique, 307-309 
density, 123 
disadvantages, 309 
effect of transducer liquid, 309 
electroding crystals, 286 

Windows, acoustic, 113-119 
design, 264-265 
function, 113 
Lucite, 116-117, 122 
materials, 116 
nylon, 117 

rubber, 114-116, 123, 264, 336-338 
steel, 117-119 
Tenite, 117 

transmission of plane waves, 113-114 
Wood’s metal, use in transducers, 121 

XCU 16257 cement, 315 
X-cut Rochelle salt crystal, use as re¬ 
search tool, 371-373 
limitations, 372-373 
radiation problems, 371 
technique, 372 
XCY8-1 transducer, 260-261 

Y-cut RS 

see Rochelle salt crystals 
Young’s modulus for crystal imped¬ 
ance, 87-91 

Z-cut ADP 

see Ammonium dihydrogen phos¬ 
phate crystals 

Zinc, use in transducers, 121 




















yj-4j 


' < . 






I V 






\ 


■ 1 •* »• 


V < 


‘-•I' 


»T '*"11 


. w, 




if 




i'.* 


\ t 


f 






Vi* 


* *r ^•• 




y > 


r •. 




-'-'V ' ■ • 


.1 A' 


« • f 








. ri 


i^’n '■ 1 








Um 




'i V 'M 

.' > I t • 


:y:':Pr■•'™ 

r ■ - . .' • C ’ i «*" 4 


■ •) ' •V-’^ 1' 


’;f .» ov, 


IT vrvi • \*i 

r’*. ^ ■ >‘t""' 

L'i ! '> t 


r*y‘ 


‘. f 


:o 


Itii 






#. 


» l.f| 




.‘♦'.,7 


ii 

' Jt .■' A* 

■ fiV 


r 

A 

1 • 

l^k ^ 

/ 

. . .';5< '* 

4* 


' ' r'T/ 


i_ 




* 


iw 


k 




:)<Ii 


,)fl 


:.':A i 






♦iU 


kf/F‘ 


3^4 • H.. ‘. ■ *' 5 

K'fc% I ^ X * I .• .% 


T>■"‘'' 

T> ? lftlCA.llMai. • ^'1* ^ 


•)^ 




fj 




. 4 J 




' I ‘V ‘ ' ■ 

HDMrf '• fir^V ^ 

' •■ ■ ■*<•' > ’’‘V ; 


/<!■ 


k * j 


k-tr. 


< \' 




V f 


«i t' 




f* . i ' 


'V • 


l?-i 


^‘J 




V 


\; 


vr ■" 


4M ‘ 


>' 


.ir * 


L'n ' 


^ i 




>. VAv 


ir 


# ■SAli 

^ 'r-’V:'.?V'.; 




^■l . ^4i 

mm 


• 'ff 


ur 


IV r 


\ifV 


N> 




.«o 

, 4.>« •»'**ry*' 

t' 


' I 


‘i> 


ii. - I 


I •* .< 


I •> «' 


•• 


N^: 






1 




• s 


9 






hh , 


r«* 


'. ) 


K 




V 


/ L 




—4 .:V.: ; ^ 


.4 *“1 


•I ► 


’%3 

Z ;,•,. ,• < ,*, 1 . ' 

^-^'■’V.-KyJltv' 

^ •w vX' 

' 7 m r -NJC <• 


••4Vi ’ ■ ,* 


' IV 








tf. 


iF\ 


’'I' 

1: 








M 
















































































































































































































































































































































































